9 39

9_38

9_40

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 39's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 39 at Knotilus!

Knot presentations

Planar diagram presentation	X1627 X3,11,4,10 X7,18,8,1 X17,13,18,12 X9,17,10,16 X5,15,6,14 X15,5,16,4 X11,3,12,2 X13,9,14,8
Gauss code	-1, 8, -2, 7, -6, 1, -3, 9, -5, 2, -8, 4, -9, 6, -7, 5, -4, 3
Dowker-Thistlethwaite code	6 10 14 18 16 2 8 4 12
Conway Notation	[2:2:20]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 12, width is 5, Braid index is 5		[{11, 6}, {2, 7}, {6, 1}, {8, 3}, {5, 2}, {7, 9}, {4, 8}, {10, 5}, {9, 11}, {3, 10}, {1, 4}]

[edit Notes on presentations of 9 39]

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 39"];

In[4]:=

PD[K]

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

Out[4]=

X1627 X3,11,4,10 X7,18,8,1 X17,13,18,12 X9,17,10,16 X5,15,6,14 X15,5,16,4 X11,3,12,2 X13,9,14,8

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

6 10 14 18 16 2 8 4 12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[2:2:20]

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}(5,\{1,1,2,-1,-3,-2,1,4,3,-2,3,4\}) }[/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]=

{ 5, 12, 5 }

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index [math]\displaystyle{ \{4,6\} }[/math] Nakanishi index 1 Maximal Thurston-Bennequin number [-1][-10] Hyperbolic Volume 12.8103 A-Polynomial See Data:9 39/A-polynomial

[edit Notes for 9 39's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 1 }[/math] Topological 4 genus [math]\displaystyle{ 1 }[/math] Concordance genus [math]\displaystyle{ 2 }[/math] Rasmussen s-Invariant 2

[edit Notes for 9 39's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -3 t^2+14 t-21+14 t^{-1} -3 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ -3 z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 55, 2 }
Jones polynomial	[math]\displaystyle{ -q^8+3 q^7-6 q^6+8 q^5-9 q^4+10 q^3-8 q^2+6 q-3+ q^{-1} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^4 a^{-2} -2 z^4 a^{-4} +z^2 a^{-2} -3 z^2 a^{-4} +3 z^2 a^{-6} +z^2+2 a^{-2} -2 a^{-4} +2 a^{-6} - a^{-8} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 z^8 a^{-4} +2 z^8 a^{-6} +5 z^7 a^{-3} +9 z^7 a^{-5} +4 z^7 a^{-7} +5 z^6 a^{-2} +5 z^6 a^{-4} +3 z^6 a^{-6} +3 z^6 a^{-8} +3 z^5 a^{-1} -7 z^5 a^{-3} -18 z^5 a^{-5} -7 z^5 a^{-7} +z^5 a^{-9} -7 z^4 a^{-2} -15 z^4 a^{-4} -13 z^4 a^{-6} -6 z^4 a^{-8} +z^4-3 z^3 a^{-1} +5 z^3 a^{-3} +12 z^3 a^{-5} +2 z^3 a^{-7} -2 z^3 a^{-9} +5 z^2 a^{-2} +12 z^2 a^{-4} +9 z^2 a^{-6} +3 z^2 a^{-8} -z^2-z a^{-3} -3 z a^{-5} -z a^{-7} +z a^{-9} -2 a^{-2} -2 a^{-4} -2 a^{-6} - a^{-8} }[/math]
The A2 invariant	[math]\displaystyle{ q^4-q^2-1+3 q^{-2} - q^{-4} +2 q^{-6} + q^{-8} - q^{-10} + q^{-12} -2 q^{-14} +2 q^{-16} - q^{-20} +2 q^{-22} - q^{-24} - q^{-26} }[/math]
The G2 invariant	[math]\displaystyle{ q^{18}-2 q^{16}+4 q^{14}-6 q^{12}+5 q^{10}-3 q^8-2 q^6+12 q^4-19 q^2+28-30 q^{-2} +21 q^{-4} -3 q^{-6} -27 q^{-8} +58 q^{-10} -76 q^{-12} +73 q^{-14} -45 q^{-16} -6 q^{-18} +63 q^{-20} -97 q^{-22} +101 q^{-24} -61 q^{-26} +2 q^{-28} +53 q^{-30} -80 q^{-32} +65 q^{-34} -12 q^{-36} -45 q^{-38} +87 q^{-40} -83 q^{-42} +36 q^{-44} +37 q^{-46} -103 q^{-48} +134 q^{-50} -123 q^{-52} +66 q^{-54} +10 q^{-56} -84 q^{-58} +131 q^{-60} -134 q^{-62} +95 q^{-64} -29 q^{-66} -43 q^{-68} +87 q^{-70} -93 q^{-72} +59 q^{-74} -52 q^{-78} +80 q^{-80} -61 q^{-82} +8 q^{-84} +57 q^{-86} -100 q^{-88} +103 q^{-90} -65 q^{-92} - q^{-94} +60 q^{-96} -93 q^{-98} +95 q^{-100} -63 q^{-102} +19 q^{-104} +19 q^{-106} -45 q^{-108} +45 q^{-110} -33 q^{-112} +17 q^{-114} -3 q^{-116} -6 q^{-118} +8 q^{-120} -8 q^{-122} +5 q^{-124} -2 q^{-126} + q^{-128} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_39.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^3-2 q+3 q^{-1} -2 q^{-3} +2 q^{-5} + q^{-7} - q^{-9} +2 q^{-11} -3 q^{-13} +2 q^{-15} - q^{-17} }[/math]
2	[math]\displaystyle{ q^{10}-2 q^8+6 q^4-8 q^2-3+18 q^{-2} -10 q^{-4} -13 q^{-6} +21 q^{-8} -2 q^{-10} -15 q^{-12} +11 q^{-14} +7 q^{-16} -7 q^{-18} -6 q^{-20} +11 q^{-22} +2 q^{-24} -18 q^{-26} +10 q^{-28} +12 q^{-30} -20 q^{-32} +2 q^{-34} +16 q^{-36} -11 q^{-38} -5 q^{-40} +8 q^{-42} - q^{-44} -2 q^{-46} + q^{-48} }[/math]
3	[math]\displaystyle{ q^{21}-2 q^{19}+3 q^{15}-7 q^{11}-q^9+19 q^7+4 q^5-35 q^3-18 q+50 q^{-1} +49 q^{-3} -58 q^{-5} -81 q^{-7} +48 q^{-9} +110 q^{-11} -22 q^{-13} -124 q^{-15} -8 q^{-17} +120 q^{-19} +35 q^{-21} -92 q^{-23} -56 q^{-25} +63 q^{-27} +66 q^{-29} -28 q^{-31} -69 q^{-33} -7 q^{-35} +70 q^{-37} +34 q^{-39} -68 q^{-41} -65 q^{-43} +66 q^{-45} +87 q^{-47} -51 q^{-49} -109 q^{-51} +28 q^{-53} +119 q^{-55} +3 q^{-57} -116 q^{-59} -33 q^{-61} +92 q^{-63} +58 q^{-65} -59 q^{-67} -66 q^{-69} +28 q^{-71} +54 q^{-73} -35 q^{-77} -11 q^{-79} +17 q^{-81} +8 q^{-83} -5 q^{-85} -4 q^{-87} + q^{-89} +2 q^{-91} - q^{-93} }[/math]
4	[math]\displaystyle{ q^{36}-2 q^{34}+3 q^{30}-3 q^{28}+q^{26}-5 q^{24}+7 q^{22}+16 q^{20}-16 q^{18}-18 q^{16}-29 q^{14}+34 q^{12}+94 q^{10}+4 q^8-88 q^6-177 q^4-4 q^2+269+223 q^{-2} -46 q^{-4} -458 q^{-6} -324 q^{-8} +281 q^{-10} +587 q^{-12} +333 q^{-14} -510 q^{-16} -757 q^{-18} -93 q^{-20} +656 q^{-22} +779 q^{-24} -153 q^{-26} -823 q^{-28} -512 q^{-30} +308 q^{-32} +823 q^{-34} +256 q^{-36} -471 q^{-38} -598 q^{-40} -94 q^{-42} +514 q^{-44} +420 q^{-46} -67 q^{-48} -446 q^{-50} -319 q^{-52} +170 q^{-54} +444 q^{-56} +222 q^{-58} -308 q^{-60} -471 q^{-62} -86 q^{-64} +492 q^{-66} +480 q^{-68} -189 q^{-70} -631 q^{-72} -374 q^{-74} +475 q^{-76} +743 q^{-78} +71 q^{-80} -654 q^{-82} -709 q^{-84} +199 q^{-86} +807 q^{-88} +460 q^{-90} -333 q^{-92} -826 q^{-94} -250 q^{-96} +475 q^{-98} +623 q^{-100} +160 q^{-102} -511 q^{-104} -453 q^{-106} -18 q^{-108} +365 q^{-110} +359 q^{-112} -71 q^{-114} -252 q^{-116} -199 q^{-118} +31 q^{-120} +191 q^{-122} +81 q^{-124} -19 q^{-126} -92 q^{-128} -50 q^{-130} +31 q^{-132} +27 q^{-134} +19 q^{-136} -11 q^{-138} -14 q^{-140} +2 q^{-142} + q^{-144} +4 q^{-146} - q^{-148} -2 q^{-150} + q^{-152} }[/math]
5	[math]\displaystyle{ q^{55}-2 q^{53}+3 q^{49}-3 q^{47}-2 q^{45}+3 q^{43}+3 q^{41}+4 q^{39}+3 q^{37}-16 q^{35}-28 q^{33}+q^{31}+45 q^{29}+67 q^{27}+26 q^{25}-78 q^{23}-176 q^{21}-133 q^{19}+107 q^{17}+362 q^{15}+373 q^{13}-4 q^{11}-574 q^9-844 q^7-376 q^5+690 q^3+1486 q+1143 q^{-1} -420 q^{-3} -2090 q^{-5} -2322 q^{-7} -436 q^{-9} +2359 q^{-11} +3622 q^{-13} +1876 q^{-15} -1907 q^{-17} -4647 q^{-19} -3684 q^{-21} +702 q^{-23} +4997 q^{-25} +5332 q^{-27} +1068 q^{-29} -4429 q^{-31} -6381 q^{-33} -2940 q^{-35} +3094 q^{-37} +6539 q^{-39} +4408 q^{-41} -1367 q^{-43} -5793 q^{-45} -5162 q^{-47} -316 q^{-49} +4443 q^{-51} +5172 q^{-53} +1581 q^{-55} -2899 q^{-57} -4539 q^{-59} -2348 q^{-61} +1461 q^{-63} +3655 q^{-65} +2656 q^{-67} -365 q^{-69} -2756 q^{-71} -2693 q^{-73} -426 q^{-75} +2053 q^{-77} +2705 q^{-79} +986 q^{-81} -1633 q^{-83} -2831 q^{-85} -1472 q^{-87} +1364 q^{-89} +3173 q^{-91} +2078 q^{-93} -1183 q^{-95} -3652 q^{-97} -2847 q^{-99} +821 q^{-101} +4136 q^{-103} +3841 q^{-105} -172 q^{-107} -4404 q^{-109} -4877 q^{-111} -889 q^{-113} +4209 q^{-115} +5774 q^{-117} +2249 q^{-119} -3374 q^{-121} -6210 q^{-123} -3713 q^{-125} +1949 q^{-127} +5923 q^{-129} +4882 q^{-131} -119 q^{-133} -4818 q^{-135} -5431 q^{-137} -1681 q^{-139} +3105 q^{-141} +5089 q^{-143} +2990 q^{-145} -1136 q^{-147} -3955 q^{-149} -3519 q^{-151} -540 q^{-153} +2391 q^{-155} +3163 q^{-157} +1561 q^{-159} -845 q^{-161} -2237 q^{-163} -1830 q^{-165} -226 q^{-167} +1171 q^{-169} +1467 q^{-171} +718 q^{-173} -315 q^{-175} -884 q^{-177} -723 q^{-179} -124 q^{-181} +366 q^{-183} +470 q^{-185} +238 q^{-187} -59 q^{-189} -223 q^{-191} -183 q^{-193} -32 q^{-195} +70 q^{-197} +84 q^{-199} +39 q^{-201} -7 q^{-203} -33 q^{-205} -22 q^{-207} +3 q^{-209} +8 q^{-211} +4 q^{-213} +2 q^{-215} - q^{-217} -4 q^{-219} + q^{-221} +2 q^{-223} - q^{-225} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^4-q^2-1+3 q^{-2} - q^{-4} +2 q^{-6} + q^{-8} - q^{-10} + q^{-12} -2 q^{-14} +2 q^{-16} - q^{-20} +2 q^{-22} - q^{-24} - q^{-26} }[/math]
1,1	[math]\displaystyle{ q^{12}-4 q^{10}+10 q^8-20 q^6+38 q^4-62 q^2+98-150 q^{-2} +211 q^{-4} -270 q^{-6} +334 q^{-8} -374 q^{-10} +372 q^{-12} -316 q^{-14} +212 q^{-16} -54 q^{-18} -136 q^{-20} +344 q^{-22} -522 q^{-24} +662 q^{-26} -745 q^{-28} +754 q^{-30} -702 q^{-32} +582 q^{-34} -415 q^{-36} +218 q^{-38} -18 q^{-40} -158 q^{-42} +298 q^{-44} -386 q^{-46} +410 q^{-48} -380 q^{-50} +319 q^{-52} -248 q^{-54} +168 q^{-56} -102 q^{-58} +58 q^{-60} -28 q^{-62} +12 q^{-64} -4 q^{-66} + q^{-68} }[/math]
2,0	[math]\displaystyle{ q^{12}-q^{10}-2 q^8+2 q^6+5 q^4-q^2-10+14 q^{-4} -11 q^{-8} +3 q^{-10} +11 q^{-12} - q^{-14} -10 q^{-16} +3 q^{-18} +6 q^{-20} -3 q^{-22} +2 q^{-24} +3 q^{-26} -2 q^{-28} +8 q^{-32} -5 q^{-34} -10 q^{-36} +2 q^{-38} +9 q^{-40} -4 q^{-42} -12 q^{-44} +6 q^{-46} +9 q^{-48} -2 q^{-50} -8 q^{-52} +6 q^{-56} -3 q^{-60} - q^{-62} + q^{-64} + q^{-66} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^8-2 q^6+5 q^2-6- q^{-2} +13 q^{-4} -10 q^{-6} -3 q^{-8} +17 q^{-10} -10 q^{-12} -4 q^{-14} +12 q^{-16} -4 q^{-18} -3 q^{-20} +2 q^{-22} +5 q^{-24} -2 q^{-26} -8 q^{-28} +8 q^{-30} +4 q^{-32} -15 q^{-34} +9 q^{-36} +7 q^{-38} -15 q^{-40} +7 q^{-42} +3 q^{-44} -8 q^{-46} +4 q^{-48} + q^{-50} -2 q^{-52} + q^{-54} }[/math]
1,0,0	[math]\displaystyle{ q^5-q^3- q^{-1} +3 q^{-3} - q^{-5} +3 q^{-7} + q^{-9} + q^{-11} - q^{-13} - q^{-15} -2 q^{-19} +2 q^{-21} +2 q^{-25} - q^{-27} +2 q^{-29} - q^{-31} - q^{-33} - q^{-35} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^8-2 q^6+4 q^4-7 q^2+10-13 q^{-2} +17 q^{-4} -16 q^{-6} +15 q^{-8} -9 q^{-10} +4 q^{-12} +4 q^{-14} -12 q^{-16} +20 q^{-18} -27 q^{-20} +30 q^{-22} -31 q^{-24} +28 q^{-26} -22 q^{-28} +16 q^{-30} -6 q^{-32} - q^{-34} +9 q^{-36} -13 q^{-38} +15 q^{-40} -17 q^{-42} +15 q^{-44} -12 q^{-46} +8 q^{-48} -5 q^{-50} +2 q^{-52} - q^{-54} }[/math]
1,0	[math]\displaystyle{ q^{14}-2 q^{10}-2 q^8+2 q^6+6 q^4+q^2-8-7 q^{-2} +6 q^{-4} +15 q^{-6} + q^{-8} -16 q^{-10} -9 q^{-12} +13 q^{-14} +15 q^{-16} -5 q^{-18} -16 q^{-20} +15 q^{-24} +5 q^{-26} -12 q^{-28} -6 q^{-30} +10 q^{-32} +9 q^{-34} -6 q^{-36} -10 q^{-38} +4 q^{-40} +11 q^{-42} -2 q^{-44} -12 q^{-46} -2 q^{-48} +13 q^{-50} +6 q^{-52} -13 q^{-54} -14 q^{-56} +9 q^{-58} +18 q^{-60} - q^{-62} -17 q^{-64} -8 q^{-66} +12 q^{-68} +11 q^{-70} -5 q^{-72} -10 q^{-74} - q^{-76} +6 q^{-78} +3 q^{-80} -2 q^{-82} -2 q^{-84} + q^{-88} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{18}-2 q^{16}+4 q^{14}-6 q^{12}+5 q^{10}-3 q^8-2 q^6+12 q^4-19 q^2+28-30 q^{-2} +21 q^{-4} -3 q^{-6} -27 q^{-8} +58 q^{-10} -76 q^{-12} +73 q^{-14} -45 q^{-16} -6 q^{-18} +63 q^{-20} -97 q^{-22} +101 q^{-24} -61 q^{-26} +2 q^{-28} +53 q^{-30} -80 q^{-32} +65 q^{-34} -12 q^{-36} -45 q^{-38} +87 q^{-40} -83 q^{-42} +36 q^{-44} +37 q^{-46} -103 q^{-48} +134 q^{-50} -123 q^{-52} +66 q^{-54} +10 q^{-56} -84 q^{-58} +131 q^{-60} -134 q^{-62} +95 q^{-64} -29 q^{-66} -43 q^{-68} +87 q^{-70} -93 q^{-72} +59 q^{-74} -52 q^{-78} +80 q^{-80} -61 q^{-82} +8 q^{-84} +57 q^{-86} -100 q^{-88} +103 q^{-90} -65 q^{-92} - q^{-94} +60 q^{-96} -93 q^{-98} +95 q^{-100} -63 q^{-102} +19 q^{-104} +19 q^{-106} -45 q^{-108} +45 q^{-110} -33 q^{-112} +17 q^{-114} -3 q^{-116} -6 q^{-118} +8 q^{-120} -8 q^{-122} +5 q^{-124} -2 q^{-126} + q^{-128} }[/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 39"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -3 t^2+14 t-21+14 t^{-1} -3 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -3 z^4+2 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]=

{ 55, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 2 z^8 a^{-4} +2 z^8 a^{-6} +5 z^7 a^{-3} +9 z^7 a^{-5} +4 z^7 a^{-7} +5 z^6 a^{-2} +5 z^6 a^{-4} +3 z^6 a^{-6} +3 z^6 a^{-8} +3 z^5 a^{-1} -7 z^5 a^{-3} -18 z^5 a^{-5} -7 z^5 a^{-7} +z^5 a^{-9} -7 z^4 a^{-2} -15 z^4 a^{-4} -13 z^4 a^{-6} -6 z^4 a^{-8} +z^4-3 z^3 a^{-1} +5 z^3 a^{-3} +12 z^3 a^{-5} +2 z^3 a^{-7} -2 z^3 a^{-9} +5 z^2 a^{-2} +12 z^2 a^{-4} +9 z^2 a^{-6} +3 z^2 a^{-8} -z^2-z a^{-3} -3 z a^{-5} -z a^{-7} +z a^{-9} -2 a^{-2} -2 a^{-4} -2 a^{-6} - a^{-8} }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n162,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11n11, K11n112,}

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 39"];

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{ -3 t^2+14 t-21+14 t^{-1} -3 t^{-2} }[/math], [math]\displaystyle{ -q^8+3 q^7-6 q^6+8 q^5-9 q^4+10 q^3-8 q^2+6 q-3+ q^{-1} }[/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]=

{K11n162,}

In[6]:=

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

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

Out[6]=

{K11n11, K11n112,}

Vassiliev invariants

V2 and V3:

(2, 4)

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{ 8 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{460}{3} }[/math] [math]\displaystyle{ \frac{116}{3} }[/math] [math]\displaystyle{ 256 }[/math] [math]\displaystyle{ \frac{2144}{3} }[/math] [math]\displaystyle{ \frac{320}{3} }[/math] [math]\displaystyle{ 192 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 512 }[/math] [math]\displaystyle{ \frac{3680}{3} }[/math] [math]\displaystyle{ \frac{928}{3} }[/math] [math]\displaystyle{ \frac{49111}{15} }[/math] [math]\displaystyle{ -\frac{8524}{15} }[/math] [math]\displaystyle{ \frac{97084}{45} }[/math] [math]\displaystyle{ \frac{857}{9} }[/math] [math]\displaystyle{ \frac{5191}{15} }[/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]2 is the signature of 9 39. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -2 -1 0 1 2 3 4 5 6 7 χ 17                   1 -1 15                 2   2 13               4 1   -3 11             4 2     2 9           5 4       -1 7         5 4         1 5       3 5           2 3     3 5             -2 1   1 4               3 -1   2                 -2 -3 1                   1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ i=3 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=7 }[/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^{23}-3 q^{22}+q^{21}+10 q^{20}-16 q^{19}-5 q^{18}+37 q^{17}-30 q^{16}-27 q^{15}+69 q^{14}-32 q^{13}-55 q^{12}+89 q^{11}-23 q^{10}-72 q^9+88 q^8-9 q^7-68 q^6+62 q^5+4 q^4-45 q^3+28 q^2+7 q-17+7 q^{-1} +2 q^{-2} -3 q^{-3} + q^{-4} }[/math]
3	[math]\displaystyle{ -q^{45}+3 q^{44}-q^{43}-5 q^{42}-2 q^{41}+16 q^{40}+8 q^{39}-33 q^{38}-26 q^{37}+51 q^{36}+62 q^{35}-59 q^{34}-120 q^{33}+58 q^{32}+179 q^{31}-25 q^{30}-245 q^{29}-25 q^{28}+298 q^{27}+91 q^{26}-336 q^{25}-162 q^{24}+356 q^{23}+229 q^{22}-357 q^{21}-293 q^{20}+353 q^{19}+331 q^{18}-321 q^{17}-370 q^{16}+291 q^{15}+372 q^{14}-227 q^{13}-373 q^{12}+172 q^{11}+336 q^{10}-100 q^9-288 q^8+44 q^7+220 q^6+2 q^5-156 q^4-18 q^3+91 q^2+25 q-49-17 q^{-1} +23 q^{-2} +8 q^{-3} -10 q^{-4} -2 q^{-5} +3 q^{-6} +2 q^{-7} -3 q^{-8} + q^{-9} }[/math]
4	[math]\displaystyle{ q^{74}-3 q^{73}+q^{72}+5 q^{71}-3 q^{70}+2 q^{69}-19 q^{68}+4 q^{67}+35 q^{66}+5 q^{65}+6 q^{64}-100 q^{63}-38 q^{62}+108 q^{61}+105 q^{60}+116 q^{59}-260 q^{58}-268 q^{57}+55 q^{56}+286 q^{55}+546 q^{54}-254 q^{53}-651 q^{52}-380 q^{51}+228 q^{50}+1217 q^{49}+209 q^{48}-799 q^{47}-1105 q^{46}-348 q^{45}+1710 q^{44}+1002 q^{43}-452 q^{42}-1713 q^{41}-1256 q^{40}+1765 q^{39}+1727 q^{38}+220 q^{37}-1981 q^{36}-2105 q^{35}+1508 q^{34}+2169 q^{33}+889 q^{32}-1969 q^{31}-2683 q^{30}+1123 q^{29}+2332 q^{28}+1419 q^{27}-1747 q^{26}-2957 q^{25}+634 q^{24}+2205 q^{23}+1798 q^{22}-1260 q^{21}-2863 q^{20}+26 q^{19}+1701 q^{18}+1925 q^{17}-533 q^{16}-2296 q^{15}-489 q^{14}+881 q^{13}+1614 q^{12}+137 q^{11}-1364 q^{10}-612 q^9+132 q^8+950 q^7+384 q^6-521 q^5-358 q^4-174 q^3+345 q^2+250 q-109-89 q^{-1} -128 q^{-2} +72 q^{-3} +77 q^{-4} -20 q^{-5} +3 q^{-6} -38 q^{-7} +12 q^{-8} +14 q^{-9} -9 q^{-10} +5 q^{-11} -6 q^{-12} +3 q^{-13} +2 q^{-14} -3 q^{-15} + q^{-16} }[/math]
5	[math]\displaystyle{ -q^{110}+3 q^{109}-q^{108}-5 q^{107}+3 q^{106}+3 q^{105}+q^{104}+7 q^{103}-6 q^{102}-30 q^{101}-8 q^{100}+29 q^{99}+47 q^{98}+52 q^{97}-20 q^{96}-132 q^{95}-159 q^{94}-11 q^{93}+211 q^{92}+349 q^{91}+212 q^{90}-236 q^{89}-649 q^{88}-610 q^{87}+50 q^{86}+918 q^{85}+1245 q^{84}+513 q^{83}-945 q^{82}-2007 q^{81}-1554 q^{80}+511 q^{79}+2637 q^{78}+2919 q^{77}+657 q^{76}-2779 q^{75}-4485 q^{74}-2468 q^{73}+2201 q^{72}+5738 q^{71}+4783 q^{70}-680 q^{69}-6469 q^{68}-7254 q^{67}-1549 q^{66}+6351 q^{65}+9482 q^{64}+4321 q^{63}-5428 q^{62}-11228 q^{61}-7211 q^{60}+3854 q^{59}+12318 q^{58}+9944 q^{57}-1903 q^{56}-12793 q^{55}-12309 q^{54}-134 q^{53}+12791 q^{52}+14176 q^{51}+2110 q^{50}-12498 q^{49}-15624 q^{48}-3802 q^{47}+11986 q^{46}+16645 q^{45}+5371 q^{44}-11403 q^{43}-17433 q^{42}-6638 q^{41}+10627 q^{40}+17843 q^{39}+7990 q^{38}-9684 q^{37}-18085 q^{36}-9117 q^{35}+8360 q^{34}+17780 q^{33}+10381 q^{32}-6663 q^{31}-17086 q^{30}-11311 q^{29}+4551 q^{28}+15589 q^{27}+12021 q^{26}-2183 q^{25}-13495 q^{24}-12040 q^{23}-208 q^{22}+10743 q^{21}+11390 q^{20}+2243 q^{19}-7720 q^{18}-9909 q^{17}-3653 q^{16}+4709 q^{15}+7949 q^{14}+4195 q^{13}-2223 q^{12}-5645 q^{11}-3988 q^{10}+414 q^9+3563 q^8+3232 q^7+517 q^6-1862 q^5-2242 q^4-849 q^3+768 q^2+1346 q+749-192 q^{-1} -679 q^{-2} -506 q^{-3} -28 q^{-4} +280 q^{-5} +281 q^{-6} +78 q^{-7} -109 q^{-8} -129 q^{-9} -39 q^{-10} +25 q^{-11} +41 q^{-12} +35 q^{-13} -11 q^{-14} -25 q^{-15} +2 q^{-16} +3 q^{-17} -3 q^{-18} +6 q^{-19} + q^{-20} -6 q^{-21} +3 q^{-22} +2 q^{-23} -3 q^{-24} + q^{-25} }[/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.

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