9 16

9_15

9_17

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 16's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 16 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X12,4,13,3 X16,6,17,5 X18,8,1,7 X6,18,7,17 X10,16,11,15 X14,10,15,9 X8,14,9,13 X2,12,3,11
Gauss code	1, -9, 2, -1, 3, -5, 4, -8, 7, -6, 9, -2, 8, -7, 6, -3, 5, -4
Dowker-Thistlethwaite code	4 12 16 18 14 2 8 10 6
Conway Notation	[3,3,2+]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 10, width is 3, Braid index is 3		[{3, 10}, {2, 6}, {1, 3}, {11, 9}, {10, 8}, {9, 7}, {8, 5}, {6, 4}, {5, 2}, {4, 11}, {7, 1}]

[edit Notes on presentations of 9 16]

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.org/wiki/KnotTheory.

In[3]:=

K = Knot["9 16"];

In[4]:=

PD[K]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

X4251 X12,4,13,3 X16,6,17,5 X18,8,1,7 X6,18,7,17 X10,16,11,15 X14,10,15,9 X8,14,9,13 X2,12,3,11

In[5]:=

GaussCode[K]

Out[5]=

1, -9, 2, -1, 3, -5, 4, -8, 7, -6, 9, -2, 8, -7, 6, -3, 5, -4

In[6]:=

DTCode[K]

Out[6]=

4 12 16 18 14 2 8 10 6

(The path below may be different on your system)

In[7]:=

AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:=

ConwayNotation[K]

Out[8]=

[3,3,2+]

In[9]:=

br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]=

[math]\displaystyle{ \textrm{BR}(3,\{1,1,1,1,2,2,-1,2,2,2\}) }[/math]

In[10]:=

{First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

KnotTheory::loading: Loading precomputed data in IndianaData`.

Out[10]=

{ 3, 10, 3 }

In[11]:=

Show[BraidPlot[br]]

Out[11]=

-Graphics-

In[12]:=

Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]=

-Graphics-

In[13]:=

ap = ArcPresentation[K]

Out[13]=

ArcPresentation[{3, 10}, {2, 6}, {1, 3}, {11, 9}, {10, 8}, {9, 7}, {8, 5}, {6, 4}, {5, 2}, {4, 11}, {7, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 3 Bridge index 3 Super bridge index [math]\displaystyle{ \{4,7\} }[/math] Nakanishi index 1 Maximal Thurston-Bennequin number [5][-16] Hyperbolic Volume 9.88301 A-Polynomial See Data:9 16/A-polynomial

[edit Notes for 9 16's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 3 }[/math] Topological 4 genus [math]\displaystyle{ 3 }[/math] Concordance genus [math]\displaystyle{ 3 }[/math] Rasmussen s-Invariant -6

[edit Notes for 9 16's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 2 t^3-5 t^2+8 t-9+8 t^{-1} -5 t^{-2} +2 t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ 2 z^6+7 z^4+6 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 39, 6 }
Jones polynomial	[math]\displaystyle{ -q^{12}+3 q^{11}-5 q^{10}+6 q^9-7 q^8+6 q^7-5 q^6+4 q^5-q^4+q^3 }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{-6} +z^6 a^{-8} +5 z^4 a^{-6} +3 z^4 a^{-8} -z^4 a^{-10} +8 z^2 a^{-6} -2 z^2 a^{-10} +4 a^{-6} -3 a^{-8} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^8 a^{-8} +z^8 a^{-10} +z^7 a^{-7} +4 z^7 a^{-9} +3 z^7 a^{-11} +z^6 a^{-6} -z^6 a^{-8} +3 z^6 a^{-10} +5 z^6 a^{-12} -2 z^5 a^{-7} -8 z^5 a^{-9} -z^5 a^{-11} +5 z^5 a^{-13} -5 z^4 a^{-6} -4 z^4 a^{-8} -8 z^4 a^{-10} -6 z^4 a^{-12} +3 z^4 a^{-14} -2 z^3 a^{-7} -z^3 a^{-9} -5 z^3 a^{-11} -5 z^3 a^{-13} +z^3 a^{-15} +8 z^2 a^{-6} +6 z^2 a^{-8} +z^2 a^{-10} +2 z^2 a^{-12} -z^2 a^{-14} +4 z a^{-7} +4 z a^{-9} +2 z a^{-11} +2 z a^{-13} -4 a^{-6} -3 a^{-8} }[/math]
The A2 invariant	[math]\displaystyle{ q^{-10} +3 q^{-14} + q^{-16} +2 q^{-18} + q^{-20} -2 q^{-22} -3 q^{-26} + q^{-34} - q^{-36} }[/math]
The G2 invariant	[math]\displaystyle{ q^{-50} +3 q^{-54} -2 q^{-56} +3 q^{-58} - q^{-60} +8 q^{-64} -11 q^{-66} +16 q^{-68} -11 q^{-70} +6 q^{-72} +11 q^{-74} -21 q^{-76} +32 q^{-78} -26 q^{-80} +18 q^{-82} + q^{-84} -23 q^{-86} +33 q^{-88} -31 q^{-90} +19 q^{-92} -18 q^{-96} +22 q^{-98} -18 q^{-100} +11 q^{-104} -24 q^{-106} +22 q^{-108} -14 q^{-110} -8 q^{-112} +27 q^{-114} -40 q^{-116} +41 q^{-118} -27 q^{-120} +2 q^{-122} +22 q^{-124} -40 q^{-126} +46 q^{-128} -36 q^{-130} +16 q^{-132} +10 q^{-134} -26 q^{-136} +30 q^{-138} -20 q^{-140} +2 q^{-142} +13 q^{-144} -19 q^{-146} +13 q^{-148} - q^{-150} -15 q^{-152} +25 q^{-154} -26 q^{-156} +19 q^{-158} -4 q^{-160} -14 q^{-162} +23 q^{-164} -27 q^{-166} +24 q^{-168} -14 q^{-170} +3 q^{-172} +6 q^{-174} -13 q^{-176} +15 q^{-178} -12 q^{-180} +9 q^{-182} -2 q^{-184} - q^{-186} +2 q^{-188} -4 q^{-190} +3 q^{-192} -2 q^{-194} + q^{-196} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_16.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{-5} +3 q^{-9} - q^{-11} + q^{-13} - q^{-15} - q^{-17} + q^{-19} -2 q^{-21} +2 q^{-23} - q^{-25} }[/math]
2	[math]\displaystyle{ q^{-10} +4 q^{-16} + q^{-18} -4 q^{-20} +6 q^{-22} +3 q^{-24} -9 q^{-26} +3 q^{-28} +6 q^{-30} -9 q^{-32} - q^{-34} +6 q^{-36} -4 q^{-38} -4 q^{-40} +3 q^{-42} +3 q^{-44} -4 q^{-46} -2 q^{-48} +9 q^{-50} -3 q^{-52} -8 q^{-54} +9 q^{-56} - q^{-58} -6 q^{-60} +5 q^{-62} -2 q^{-66} + q^{-68} }[/math]
3	[math]\displaystyle{ q^{-15} + q^{-21} +4 q^{-23} + q^{-25} -3 q^{-27} -2 q^{-29} +9 q^{-31} +8 q^{-33} -7 q^{-35} -16 q^{-37} +6 q^{-39} +20 q^{-41} +2 q^{-43} -26 q^{-45} -12 q^{-47} +25 q^{-49} +18 q^{-51} -21 q^{-53} -27 q^{-55} +14 q^{-57} +28 q^{-59} -7 q^{-61} -32 q^{-63} + q^{-65} +27 q^{-67} +8 q^{-69} -26 q^{-71} -9 q^{-73} +21 q^{-75} +16 q^{-77} -13 q^{-79} -19 q^{-81} +3 q^{-83} +21 q^{-85} +8 q^{-87} -22 q^{-89} -19 q^{-91} +18 q^{-93} +29 q^{-95} -14 q^{-97} -32 q^{-99} +8 q^{-101} +31 q^{-103} -3 q^{-105} -23 q^{-107} - q^{-109} +16 q^{-111} -11 q^{-115} +2 q^{-117} +5 q^{-119} -3 q^{-123} +2 q^{-127} - q^{-129} }[/math]
4	[math]\displaystyle{ q^{-20} + q^{-26} + q^{-28} +4 q^{-30} -3 q^{-34} + q^{-38} +13 q^{-40} +6 q^{-42} -8 q^{-44} -15 q^{-46} -15 q^{-48} +22 q^{-50} +30 q^{-52} +11 q^{-54} -26 q^{-56} -60 q^{-58} -4 q^{-60} +44 q^{-62} +65 q^{-64} +16 q^{-66} -86 q^{-68} -68 q^{-70} -3 q^{-72} +93 q^{-74} +94 q^{-76} -44 q^{-78} -99 q^{-80} -84 q^{-82} +50 q^{-84} +135 q^{-86} +37 q^{-88} -67 q^{-90} -132 q^{-92} -17 q^{-94} +118 q^{-96} +88 q^{-98} -14 q^{-100} -130 q^{-102} -60 q^{-104} +80 q^{-106} +103 q^{-108} +18 q^{-110} -109 q^{-112} -75 q^{-114} +47 q^{-116} +101 q^{-118} +35 q^{-120} -77 q^{-122} -85 q^{-124} +86 q^{-128} +64 q^{-130} -17 q^{-132} -83 q^{-134} -72 q^{-136} +36 q^{-138} +91 q^{-140} +78 q^{-142} -46 q^{-144} -135 q^{-146} -46 q^{-148} +68 q^{-150} +147 q^{-152} +25 q^{-154} -129 q^{-156} -100 q^{-158} +6 q^{-160} +137 q^{-162} +67 q^{-164} -69 q^{-166} -76 q^{-168} -34 q^{-170} +71 q^{-172} +49 q^{-174} -24 q^{-176} -26 q^{-178} -25 q^{-180} +27 q^{-182} +15 q^{-184} -12 q^{-186} - q^{-188} -8 q^{-190} +10 q^{-192} +3 q^{-194} -6 q^{-196} + q^{-198} -2 q^{-200} +3 q^{-202} -2 q^{-206} + q^{-208} }[/math]
5	[math]\displaystyle{ q^{-25} + q^{-31} + q^{-33} + q^{-35} +3 q^{-37} -3 q^{-41} +4 q^{-45} +5 q^{-47} +10 q^{-49} +3 q^{-51} -11 q^{-53} -17 q^{-55} -9 q^{-57} +7 q^{-59} +33 q^{-61} +35 q^{-63} +4 q^{-65} -40 q^{-67} -67 q^{-69} -48 q^{-71} +27 q^{-73} +98 q^{-75} +102 q^{-77} +28 q^{-79} -97 q^{-81} -177 q^{-83} -115 q^{-85} +48 q^{-87} +201 q^{-89} +229 q^{-91} +69 q^{-93} -185 q^{-95} -320 q^{-97} -211 q^{-99} +71 q^{-101} +343 q^{-103} +366 q^{-105} +91 q^{-107} -288 q^{-109} -462 q^{-111} -280 q^{-113} +140 q^{-115} +481 q^{-117} +454 q^{-119} +46 q^{-121} -418 q^{-123} -555 q^{-125} -241 q^{-127} +281 q^{-129} +592 q^{-131} +400 q^{-133} -129 q^{-135} -548 q^{-137} -500 q^{-139} -21 q^{-141} +476 q^{-143} +542 q^{-145} +129 q^{-147} -374 q^{-149} -537 q^{-151} -202 q^{-153} +311 q^{-155} +495 q^{-157} +221 q^{-159} -238 q^{-161} -462 q^{-163} -233 q^{-165} +212 q^{-167} +424 q^{-169} +221 q^{-171} -174 q^{-173} -401 q^{-175} -242 q^{-177} +139 q^{-179} +380 q^{-181} +275 q^{-183} -64 q^{-185} -342 q^{-187} -333 q^{-189} -54 q^{-191} +274 q^{-193} +392 q^{-195} +209 q^{-197} -154 q^{-199} -423 q^{-201} -384 q^{-203} -23 q^{-205} +406 q^{-207} +536 q^{-209} +229 q^{-211} -307 q^{-213} -634 q^{-215} -432 q^{-217} +153 q^{-219} +637 q^{-221} +589 q^{-223} +24 q^{-225} -555 q^{-227} -648 q^{-229} -184 q^{-231} +404 q^{-233} +617 q^{-235} +289 q^{-237} -251 q^{-239} -502 q^{-241} -307 q^{-243} +105 q^{-245} +359 q^{-247} +273 q^{-249} -21 q^{-251} -229 q^{-253} -193 q^{-255} -15 q^{-257} +122 q^{-259} +119 q^{-261} +24 q^{-263} -62 q^{-265} -68 q^{-267} -6 q^{-269} +28 q^{-271} +24 q^{-273} +6 q^{-275} -13 q^{-277} -11 q^{-279} + q^{-281} +9 q^{-283} +2 q^{-285} -5 q^{-287} -2 q^{-289} + q^{-291} + q^{-295} +2 q^{-297} -3 q^{-299} +2 q^{-303} - q^{-305} }[/math]
6	[math]\displaystyle{ q^{-30} + q^{-36} + q^{-38} + q^{-40} +3 q^{-44} -3 q^{-48} + q^{-50} +4 q^{-52} +7 q^{-54} +2 q^{-56} +7 q^{-58} - q^{-60} -15 q^{-62} -14 q^{-64} -4 q^{-66} +14 q^{-68} +18 q^{-70} +39 q^{-72} +23 q^{-74} -21 q^{-76} -57 q^{-78} -70 q^{-80} -38 q^{-82} +2 q^{-84} +103 q^{-86} +142 q^{-88} +98 q^{-90} -16 q^{-92} -149 q^{-94} -226 q^{-96} -226 q^{-98} -12 q^{-100} +216 q^{-102} +378 q^{-104} +341 q^{-106} +108 q^{-108} -250 q^{-110} -594 q^{-112} -542 q^{-114} -230 q^{-116} +304 q^{-118} +733 q^{-120} +844 q^{-122} +446 q^{-124} -359 q^{-126} -939 q^{-128} -1136 q^{-130} -653 q^{-132} +242 q^{-134} +1195 q^{-136} +1504 q^{-138} +875 q^{-140} -194 q^{-142} -1363 q^{-144} -1799 q^{-146} -1273 q^{-148} +193 q^{-150} +1597 q^{-152} +2082 q^{-154} +1504 q^{-156} -118 q^{-158} -1745 q^{-160} -2514 q^{-162} -1636 q^{-164} +195 q^{-166} +1945 q^{-168} +2712 q^{-170} +1765 q^{-172} -273 q^{-174} -2326 q^{-176} -2802 q^{-178} -1647 q^{-180} +531 q^{-182} +2513 q^{-184} +2866 q^{-186} +1428 q^{-188} -1036 q^{-190} -2643 q^{-192} -2650 q^{-194} -948 q^{-196} +1406 q^{-198} +2752 q^{-200} +2262 q^{-202} +202 q^{-204} -1745 q^{-206} -2565 q^{-208} -1602 q^{-210} +424 q^{-212} +2035 q^{-214} +2152 q^{-216} +701 q^{-218} -1003 q^{-220} -1991 q^{-222} -1457 q^{-224} +73 q^{-226} +1462 q^{-228} +1680 q^{-230} +600 q^{-232} -764 q^{-234} -1576 q^{-236} -1118 q^{-238} +116 q^{-240} +1277 q^{-242} +1445 q^{-244} +522 q^{-246} -723 q^{-248} -1509 q^{-250} -1174 q^{-252} -96 q^{-254} +1125 q^{-256} +1590 q^{-258} +1040 q^{-260} -200 q^{-262} -1369 q^{-264} -1675 q^{-266} -1067 q^{-268} +310 q^{-270} +1560 q^{-272} +2010 q^{-274} +1189 q^{-276} -404 q^{-278} -1854 q^{-280} -2408 q^{-282} -1420 q^{-284} +534 q^{-286} +2418 q^{-288} +2785 q^{-290} +1487 q^{-292} -840 q^{-294} -2898 q^{-296} -3129 q^{-298} -1380 q^{-300} +1432 q^{-302} +3224 q^{-304} +3100 q^{-306} +995 q^{-308} -1852 q^{-310} -3415 q^{-312} -2780 q^{-314} -279 q^{-316} +2092 q^{-318} +3141 q^{-320} +2142 q^{-322} -230 q^{-324} -2201 q^{-326} -2606 q^{-328} -1226 q^{-330} +569 q^{-332} +1893 q^{-334} +1871 q^{-336} +601 q^{-338} -806 q^{-340} -1463 q^{-342} -1008 q^{-344} -171 q^{-346} +688 q^{-348} +960 q^{-350} +510 q^{-352} -133 q^{-354} -525 q^{-356} -424 q^{-358} -203 q^{-360} +144 q^{-362} +323 q^{-364} +193 q^{-366} -3 q^{-368} -131 q^{-370} -87 q^{-372} -73 q^{-374} +13 q^{-376} +80 q^{-378} +36 q^{-380} -9 q^{-382} -27 q^{-384} +4 q^{-386} -14 q^{-388} +16 q^{-392} -7 q^{-396} -7 q^{-398} +11 q^{-400} -3 q^{-402} -2 q^{-404} +5 q^{-406} -2 q^{-408} - q^{-410} -2 q^{-412} +3 q^{-414} -2 q^{-418} + q^{-420} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{-10} +3 q^{-14} + q^{-16} +2 q^{-18} + q^{-20} -2 q^{-22} -3 q^{-26} + q^{-34} - q^{-36} }[/math]
1,1	[math]\displaystyle{ q^{-20} +6 q^{-24} -4 q^{-26} +20 q^{-28} -22 q^{-30} +50 q^{-32} -58 q^{-34} +83 q^{-36} -98 q^{-38} +102 q^{-40} -104 q^{-42} +75 q^{-44} -60 q^{-46} +4 q^{-48} +34 q^{-50} -85 q^{-52} +128 q^{-54} -162 q^{-56} +192 q^{-58} -189 q^{-60} +184 q^{-62} -156 q^{-64} +118 q^{-66} -73 q^{-68} +24 q^{-70} +16 q^{-72} -52 q^{-74} +73 q^{-76} -84 q^{-78} +88 q^{-80} -84 q^{-82} +74 q^{-84} -58 q^{-86} +46 q^{-88} -36 q^{-90} +22 q^{-92} -12 q^{-94} +8 q^{-96} -4 q^{-98} + q^{-100} }[/math]
2,0	[math]\displaystyle{ q^{-20} +3 q^{-26} +4 q^{-28} +3 q^{-32} +6 q^{-34} +4 q^{-36} - q^{-38} +2 q^{-40} +3 q^{-42} -4 q^{-44} -6 q^{-46} - q^{-48} -5 q^{-50} -8 q^{-52} -2 q^{-54} - q^{-56} -3 q^{-58} +7 q^{-62} +4 q^{-64} +3 q^{-68} +3 q^{-70} -4 q^{-72} -4 q^{-74} + q^{-76} + q^{-84} - q^{-88} + q^{-90} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{-20} +3 q^{-24} +4 q^{-26} +2 q^{-28} +7 q^{-30} +6 q^{-32} -4 q^{-34} +4 q^{-36} -3 q^{-38} -12 q^{-40} - q^{-44} -7 q^{-46} +3 q^{-48} +5 q^{-50} - q^{-52} - q^{-54} +3 q^{-58} -5 q^{-60} -2 q^{-62} +7 q^{-64} -5 q^{-66} -3 q^{-68} +8 q^{-70} -3 q^{-72} -3 q^{-74} +5 q^{-76} - q^{-78} -2 q^{-80} + q^{-82} }[/math]
1,0,0	[math]\displaystyle{ q^{-15} +3 q^{-19} + q^{-21} +4 q^{-23} + q^{-25} +2 q^{-27} - q^{-29} -2 q^{-31} -2 q^{-33} -3 q^{-35} - q^{-39} +2 q^{-41} - q^{-43} + q^{-45} - q^{-47} }[/math]
1,0,1	[math]\displaystyle{ q^{-30} +6 q^{-34} +2 q^{-36} +11 q^{-38} +15 q^{-40} - q^{-42} +38 q^{-44} -18 q^{-46} +27 q^{-48} +10 q^{-50} -48 q^{-52} +63 q^{-54} -97 q^{-56} +36 q^{-58} -17 q^{-60} -72 q^{-62} +84 q^{-64} -102 q^{-66} +61 q^{-68} -3 q^{-70} -27 q^{-72} +56 q^{-74} -21 q^{-76} +6 q^{-78} +21 q^{-80} -35 q^{-84} +64 q^{-86} -71 q^{-88} +26 q^{-90} +30 q^{-92} -84 q^{-94} +97 q^{-96} -61 q^{-98} +2 q^{-100} +53 q^{-102} -76 q^{-104} +61 q^{-106} -21 q^{-108} -24 q^{-110} +48 q^{-112} -42 q^{-114} +19 q^{-116} +9 q^{-118} -24 q^{-120} +19 q^{-122} -7 q^{-124} -2 q^{-126} +6 q^{-128} -4 q^{-130} + q^{-132} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{-30} +3 q^{-34} +4 q^{-36} +5 q^{-38} +7 q^{-40} +12 q^{-42} +6 q^{-44} +6 q^{-46} +6 q^{-48} -3 q^{-50} -10 q^{-52} -8 q^{-54} -8 q^{-56} -15 q^{-58} -8 q^{-60} +3 q^{-62} +3 q^{-64} -3 q^{-66} +8 q^{-68} +8 q^{-70} -5 q^{-72} -2 q^{-74} +3 q^{-76} -5 q^{-78} -6 q^{-80} +3 q^{-82} +2 q^{-84} -3 q^{-86} + q^{-88} +6 q^{-90} -3 q^{-94} +3 q^{-96} +2 q^{-98} -3 q^{-100} - q^{-102} + q^{-104} }[/math]
1,0,0,0	[math]\displaystyle{ q^{-20} +3 q^{-24} + q^{-26} +4 q^{-28} +3 q^{-30} +2 q^{-32} +2 q^{-34} - q^{-36} - q^{-38} -4 q^{-40} -2 q^{-42} -3 q^{-44} - q^{-48} + q^{-50} + q^{-52} - q^{-54} + q^{-56} - q^{-58} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{-20} +3 q^{-24} -2 q^{-26} +6 q^{-28} -5 q^{-30} +8 q^{-32} -6 q^{-34} +8 q^{-36} -5 q^{-38} +2 q^{-40} -5 q^{-44} +7 q^{-46} -13 q^{-48} +13 q^{-50} -15 q^{-52} +13 q^{-54} -12 q^{-56} +9 q^{-58} -5 q^{-60} +2 q^{-62} +3 q^{-64} -5 q^{-66} +7 q^{-68} -8 q^{-70} +7 q^{-72} -7 q^{-74} +5 q^{-76} -3 q^{-78} +2 q^{-80} - q^{-82} }[/math]
1,0	[math]\displaystyle{ q^{-30} +3 q^{-38} +3 q^{-40} + q^{-42} -2 q^{-44} +2 q^{-46} +7 q^{-48} +6 q^{-50} -4 q^{-52} -5 q^{-54} + q^{-56} +8 q^{-58} + q^{-60} -10 q^{-62} -8 q^{-64} +2 q^{-66} +5 q^{-68} -3 q^{-70} -7 q^{-72} -2 q^{-74} +5 q^{-76} +2 q^{-78} -3 q^{-80} -2 q^{-82} +5 q^{-84} +4 q^{-86} -3 q^{-88} -6 q^{-90} +2 q^{-92} +6 q^{-94} - q^{-96} -7 q^{-98} -2 q^{-100} +7 q^{-102} +4 q^{-104} -5 q^{-106} -7 q^{-108} +2 q^{-110} +8 q^{-112} +2 q^{-114} -5 q^{-116} -5 q^{-118} +2 q^{-120} +5 q^{-122} + q^{-124} -2 q^{-126} -2 q^{-128} + q^{-132} }[/math]

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\displaystyle{ q^{-30} +3 q^{-34} + q^{-36} +7 q^{-38} + q^{-40} +10 q^{-42} +10 q^{-46} -4 q^{-48} +4 q^{-50} -8 q^{-52} -8 q^{-56} -6 q^{-58} -2 q^{-60} -6 q^{-62} +5 q^{-64} -8 q^{-66} +11 q^{-68} -8 q^{-70} +12 q^{-72} -11 q^{-74} +10 q^{-76} -9 q^{-78} +8 q^{-80} -7 q^{-82} +2 q^{-84} -2 q^{-86} +3 q^{-90} -5 q^{-92} +4 q^{-94} -5 q^{-96} +8 q^{-98} -5 q^{-100} +4 q^{-102} -5 q^{-104} +5 q^{-106} -2 q^{-108} + q^{-110} -2 q^{-112} + q^{-114} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{-50} +3 q^{-54} -2 q^{-56} +3 q^{-58} - q^{-60} +8 q^{-64} -11 q^{-66} +16 q^{-68} -11 q^{-70} +6 q^{-72} +11 q^{-74} -21 q^{-76} +32 q^{-78} -26 q^{-80} +18 q^{-82} + q^{-84} -23 q^{-86} +33 q^{-88} -31 q^{-90} +19 q^{-92} -18 q^{-96} +22 q^{-98} -18 q^{-100} +11 q^{-104} -24 q^{-106} +22 q^{-108} -14 q^{-110} -8 q^{-112} +27 q^{-114} -40 q^{-116} +41 q^{-118} -27 q^{-120} +2 q^{-122} +22 q^{-124} -40 q^{-126} +46 q^{-128} -36 q^{-130} +16 q^{-132} +10 q^{-134} -26 q^{-136} +30 q^{-138} -20 q^{-140} +2 q^{-142} +13 q^{-144} -19 q^{-146} +13 q^{-148} - q^{-150} -15 q^{-152} +25 q^{-154} -26 q^{-156} +19 q^{-158} -4 q^{-160} -14 q^{-162} +23 q^{-164} -27 q^{-166} +24 q^{-168} -14 q^{-170} +3 q^{-172} +6 q^{-174} -13 q^{-176} +15 q^{-178} -12 q^{-180} +9 q^{-182} -2 q^{-184} - q^{-186} +2 q^{-188} -4 q^{-190} +3 q^{-192} -2 q^{-194} + q^{-196} }[/math]

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 16"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

[math]\displaystyle{ 2 t^3-5 t^2+8 t-9+8 t^{-1} -5 t^{-2} +2 t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 2 z^6+7 z^4+6 z^2+1 }[/math]

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

[math]\displaystyle{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 39, 6 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

[math]\displaystyle{ -q^{12}+3 q^{11}-5 q^{10}+6 q^9-7 q^8+6 q^7-5 q^6+4 q^5-q^4+q^3 }[/math]

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

[math]\displaystyle{ z^6 a^{-6} +z^6 a^{-8} +5 z^4 a^{-6} +3 z^4 a^{-8} -z^4 a^{-10} +8 z^2 a^{-6} -2 z^2 a^{-10} +4 a^{-6} -3 a^{-8} }[/math]

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

[math]\displaystyle{ z^8 a^{-8} +z^8 a^{-10} +z^7 a^{-7} +4 z^7 a^{-9} +3 z^7 a^{-11} +z^6 a^{-6} -z^6 a^{-8} +3 z^6 a^{-10} +5 z^6 a^{-12} -2 z^5 a^{-7} -8 z^5 a^{-9} -z^5 a^{-11} +5 z^5 a^{-13} -5 z^4 a^{-6} -4 z^4 a^{-8} -8 z^4 a^{-10} -6 z^4 a^{-12} +3 z^4 a^{-14} -2 z^3 a^{-7} -z^3 a^{-9} -5 z^3 a^{-11} -5 z^3 a^{-13} +z^3 a^{-15} +8 z^2 a^{-6} +6 z^2 a^{-8} +z^2 a^{-10} +2 z^2 a^{-12} -z^2 a^{-14} +4 z a^{-7} +4 z a^{-9} +2 z a^{-11} +2 z a^{-13} -4 a^{-6} -3 a^{-8} }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 16"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ [math]\displaystyle{ 2 t^3-5 t^2+8 t-9+8 t^{-1} -5 t^{-2} +2 t^{-3} }[/math], [math]\displaystyle{ -q^{12}+3 q^{11}-5 q^{10}+6 q^9-7 q^8+6 q^7-5 q^6+4 q^5-q^4+q^3 }[/math] }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(6, 14)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 [math]\displaystyle{ 24 }[/math] [math]\displaystyle{ 112 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 668 }[/math] [math]\displaystyle{ 84 }[/math] [math]\displaystyle{ 2688 }[/math] [math]\displaystyle{ \frac{13408}{3} }[/math] [math]\displaystyle{ \frac{2176}{3} }[/math] [math]\displaystyle{ 496 }[/math] [math]\displaystyle{ 2304 }[/math] [math]\displaystyle{ 6272 }[/math] [math]\displaystyle{ 16032 }[/math] [math]\displaystyle{ 2016 }[/math] [math]\displaystyle{ \frac{154271}{5} }[/math] [math]\displaystyle{ \frac{23548}{15} }[/math] [math]\displaystyle{ \frac{153164}{15} }[/math] [math]\displaystyle{ \frac{769}{3} }[/math] [math]\displaystyle{ \frac{6111}{5} }[/math]

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]6 is the signature of 9 16. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 8 9 χ 25                   1 -1 23                 2   2 21               3 1   -2 19             3 2     1 17           4 3       -1 15         2 3         -1 13       3 4           1 11     1 2             -1 9     3               3 7 1 1                 0 5 1                   1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=5 }[/math] [math]\displaystyle{ i=7 }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=9 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the [math]\displaystyle{ n+1 }[/math]-dimensional representation of [math]\displaystyle{ sl(2) }[/math])

[math]\displaystyle{ n }[/math]	[math]\displaystyle{ J_n }[/math]
2	[math]\displaystyle{ q^{33}-3 q^{32}+2 q^{31}+6 q^{30}-14 q^{29}+7 q^{28}+16 q^{27}-31 q^{26}+12 q^{25}+28 q^{24}-42 q^{23}+10 q^{22}+35 q^{21}-42 q^{20}+3 q^{19}+35 q^{18}-32 q^{17}-4 q^{16}+27 q^{15}-17 q^{14}-7 q^{13}+15 q^{12}-5 q^{11}-4 q^{10}+5 q^9-q^7+q^6 }[/math]
3	[math]\displaystyle{ -q^{63}+3 q^{62}-2 q^{61}-3 q^{60}+2 q^{59}+8 q^{58}-5 q^{57}-16 q^{56}+13 q^{55}+24 q^{54}-22 q^{53}-38 q^{52}+33 q^{51}+58 q^{50}-45 q^{49}-78 q^{48}+51 q^{47}+101 q^{46}-56 q^{45}-115 q^{44}+48 q^{43}+131 q^{42}-43 q^{41}-133 q^{40}+26 q^{39}+137 q^{38}-14 q^{37}-128 q^{36}-4 q^{35}+120 q^{34}+20 q^{33}-109 q^{32}-30 q^{31}+87 q^{30}+45 q^{29}-74 q^{28}-44 q^{27}+46 q^{26}+51 q^{25}-35 q^{24}-37 q^{23}+9 q^{22}+37 q^{21}-7 q^{20}-19 q^{19}-5 q^{18}+15 q^{17}+2 q^{16}-4 q^{15}-4 q^{14}+4 q^{13}+q^{12}-q^{10}+q^9 }[/math]
4	[math]\displaystyle{ q^{102}-3 q^{101}+2 q^{100}+3 q^{99}-5 q^{98}+4 q^{97}-10 q^{96}+11 q^{95}+10 q^{94}-23 q^{93}+11 q^{92}-21 q^{91}+38 q^{90}+22 q^{89}-75 q^{88}+10 q^{87}-19 q^{86}+111 q^{85}+44 q^{84}-180 q^{83}-32 q^{82}-12 q^{81}+247 q^{80}+114 q^{79}-311 q^{78}-138 q^{77}-41 q^{76}+401 q^{75}+236 q^{74}-390 q^{73}-252 q^{72}-130 q^{71}+490 q^{70}+360 q^{69}-377 q^{68}-307 q^{67}-238 q^{66}+479 q^{65}+426 q^{64}-296 q^{63}-285 q^{62}-324 q^{61}+394 q^{60}+434 q^{59}-184 q^{58}-219 q^{57}-378 q^{56}+272 q^{55}+400 q^{54}-57 q^{53}-134 q^{52}-401 q^{51}+132 q^{50}+330 q^{49}+59 q^{48}-32 q^{47}-371 q^{46}-3 q^{45}+215 q^{44}+124 q^{43}+72 q^{42}-273 q^{41}-88 q^{40}+81 q^{39}+109 q^{38}+127 q^{37}-135 q^{36}-89 q^{35}-15 q^{34}+44 q^{33}+109 q^{32}-33 q^{31}-40 q^{30}-36 q^{29}-4 q^{28}+53 q^{27}+q^{26}-3 q^{25}-17 q^{24}-12 q^{23}+16 q^{22}+q^{21}+4 q^{20}-3 q^{19}-5 q^{18}+4 q^{17}+q^{15}-q^{13}+q^{12} }[/math]
5	[math]\displaystyle{ -q^{150}+3 q^{149}-2 q^{148}-3 q^{147}+5 q^{146}-q^{145}-2 q^{144}+4 q^{143}-5 q^{142}-6 q^{141}+12 q^{140}+6 q^{139}-10 q^{138}-8 q^{137}-7 q^{136}+13 q^{135}+30 q^{134}+10 q^{133}-44 q^{132}-70 q^{131}-q^{130}+99 q^{129}+125 q^{128}+13 q^{127}-181 q^{126}-248 q^{125}-37 q^{124}+307 q^{123}+419 q^{122}+99 q^{121}-435 q^{120}-660 q^{119}-232 q^{118}+558 q^{117}+959 q^{116}+427 q^{115}-648 q^{114}-1248 q^{113}-696 q^{112}+651 q^{111}+1538 q^{110}+992 q^{109}-600 q^{108}-1732 q^{107}-1281 q^{106}+449 q^{105}+1865 q^{104}+1528 q^{103}-293 q^{102}-1862 q^{101}-1710 q^{100}+88 q^{99}+1826 q^{98}+1797 q^{97}+70 q^{96}-1679 q^{95}-1828 q^{94}-240 q^{93}+1547 q^{92}+1788 q^{91}+348 q^{90}-1340 q^{89}-1723 q^{88}-481 q^{87}+1166 q^{86}+1629 q^{85}+575 q^{84}-945 q^{83}-1520 q^{82}-693 q^{81}+721 q^{80}+1400 q^{79}+799 q^{78}-486 q^{77}-1246 q^{76}-877 q^{75}+208 q^{74}+1065 q^{73}+962 q^{72}+17 q^{71}-833 q^{70}-943 q^{69}-289 q^{68}+586 q^{67}+914 q^{66}+436 q^{65}-304 q^{64}-751 q^{63}-600 q^{62}+64 q^{61}+600 q^{60}+573 q^{59}+160 q^{58}-343 q^{57}-573 q^{56}-277 q^{55}+180 q^{54}+391 q^{53}+334 q^{52}+36 q^{51}-298 q^{50}-300 q^{49}-92 q^{48}+109 q^{47}+225 q^{46}+171 q^{45}-44 q^{44}-140 q^{43}-120 q^{42}-44 q^{41}+62 q^{40}+109 q^{39}+36 q^{38}-15 q^{37}-46 q^{36}-48 q^{35}-9 q^{34}+34 q^{33}+17 q^{32}+12 q^{31}-2 q^{30}-17 q^{29}-11 q^{28}+8 q^{27}+q^{26}+4 q^{25}+4 q^{24}-3 q^{23}-4 q^{22}+3 q^{21}+q^{18}-q^{16}+q^{15} }[/math]
6	[math]\displaystyle{ q^{207}-3 q^{206}+2 q^{205}+3 q^{204}-5 q^{203}+q^{202}-q^{201}+8 q^{200}-10 q^{199}+q^{198}+17 q^{197}-23 q^{196}+q^{195}+6 q^{194}+24 q^{193}-26 q^{192}-13 q^{191}+35 q^{190}-54 q^{189}+19 q^{188}+51 q^{187}+68 q^{186}-93 q^{185}-99 q^{184}+21 q^{183}-98 q^{182}+147 q^{181}+247 q^{180}+198 q^{179}-272 q^{178}-446 q^{177}-200 q^{176}-199 q^{175}+539 q^{174}+890 q^{173}+648 q^{172}-544 q^{171}-1305 q^{170}-1037 q^{169}-654 q^{168}+1196 q^{167}+2297 q^{166}+1918 q^{165}-522 q^{164}-2629 q^{163}-2832 q^{162}-2034 q^{161}+1601 q^{160}+4268 q^{159}+4290 q^{158}+477 q^{157}-3678 q^{156}-5203 q^{155}-4535 q^{154}+966 q^{153}+5831 q^{152}+7137 q^{151}+2582 q^{150}-3554 q^{149}-6995 q^{148}-7347 q^{147}-783 q^{146}+6062 q^{145}+9195 q^{144}+4909 q^{143}-2256 q^{142}-7362 q^{141}-9231 q^{140}-2737 q^{139}+5074 q^{138}+9749 q^{137}+6359 q^{136}-663 q^{135}-6541 q^{134}-9681 q^{133}-3987 q^{132}+3697 q^{131}+9141 q^{130}+6665 q^{129}+506 q^{128}-5301 q^{127}-9131 q^{126}-4452 q^{125}+2476 q^{124}+8063 q^{123}+6330 q^{122}+1292 q^{121}-4056 q^{120}-8208 q^{119}-4620 q^{118}+1315 q^{117}+6829 q^{116}+5872 q^{115}+2104 q^{114}-2692 q^{113}-7128 q^{112}-4838 q^{111}-74 q^{110}+5299 q^{109}+5338 q^{108}+3092 q^{107}-988 q^{106}-5677 q^{105}-4955 q^{104}-1685 q^{103}+3273 q^{102}+4375 q^{101}+3912 q^{100}+959 q^{99}-3617 q^{98}-4465 q^{97}-3031 q^{96}+919 q^{95}+2673 q^{94}+3919 q^{93}+2566 q^{92}-1153 q^{91}-3027 q^{90}-3384 q^{89}-1063 q^{88}+495 q^{87}+2764 q^{86}+3042 q^{85}+900 q^{84}-989 q^{83}-2437 q^{82}-1830 q^{81}-1255 q^{80}+933 q^{79}+2164 q^{78}+1669 q^{77}+638 q^{76}-815 q^{75}-1252 q^{74}-1740 q^{73}-471 q^{72}+698 q^{71}+1143 q^{70}+1074 q^{69}+354 q^{68}-183 q^{67}-1111 q^{66}-780 q^{65}-255 q^{64}+248 q^{63}+591 q^{62}+551 q^{61}+397 q^{60}-306 q^{59}-382 q^{58}-366 q^{57}-181 q^{56}+57 q^{55}+239 q^{54}+345 q^{53}+38 q^{52}-24 q^{51}-133 q^{50}-144 q^{49}-105 q^{48}+11 q^{47}+131 q^{46}+38 q^{45}+53 q^{44}-30 q^{42}-61 q^{41}-28 q^{40}+30 q^{39}-2 q^{38}+21 q^{37}+13 q^{36}+6 q^{35}-17 q^{34}-12 q^{33}+9 q^{32}-6 q^{31}+3 q^{30}+3 q^{29}+5 q^{28}-3 q^{27}-4 q^{26}+4 q^{25}-q^{24}+q^{21}-q^{19}+q^{18} }[/math]
7	[math]\displaystyle{ \textrm{NotAvailable}(q) }[/math]

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top.

9_15 9_17