9 39

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9 38.gif

9_38

9 40.gif

9_40

9 39.gif Visit 9 39's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 9 39's page at Knotilus!

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9 39 Quick Notes


9 39 Further Notes and Views

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]

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]

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]

V2,1 through V6,9 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 \
 -2-101234567χ
17         1-1
15        2 2
13       41 -3
11      42  2
9     54   -1
7    54    1
5   35     2
3  35      -2
1 14       3
-1 2        -2
-31         1

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

[math]\displaystyle{ \textrm{Include}(\textrm{ColouredJonesM.mhtml}) }[/math] 

In[1]:=    
 
<< KnotTheory`
 
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=
Crossings[Knot[9, 39]]
Out[2]=  
9
In[3]:=
PD[Knot[9, 39]]
Out[3]=  
PD[X[1, 6, 2, 7], X[3, 11, 4, 10], X[7, 18, 8, 1], X[17, 13, 18, 12], 

 X[9, 17, 10, 16], X[5, 15, 6, 14], X[15, 5, 16, 4], X[11, 3, 12, 2], 



  X[13, 9, 14, 8]]
In[4]:=
GaussCode[Knot[9, 39]]
Out[4]=  
GaussCode[-1, 8, -2, 7, -6, 1, -3, 9, -5, 2, -8, 4, -9, 6, -7, 5, -4, 3]
In[5]:=
BR[Knot[9, 39]]
Out[5]=  
BR[5, {1, 1, 2, -1, -3, -2, 1, 4, 3, -2, 3, 4}]
In[6]:=
alex = Alexander[Knot[9, 39]][t]
Out[6]=  
      3    14             2

-21 - -- + -- + 14 t - 3 t



      2   t


      t
In[7]:=
Conway[Knot[9, 39]][z]
Out[7]=  
       2      4
1 + 2 z  - 3 z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[9, 39], Knot[11, NonAlternating, 162]}
In[9]:=
{KnotDet[Knot[9, 39]], KnotSignature[Knot[9, 39]]}
Out[9]=  
{55, 2}
In[10]:=
J=Jones[Knot[9, 39]][q]
Out[10]=  
     1            2       3      4      5      6      7    8

-3 + - + 6 q - 8 q  + 10 q  - 9 q  + 8 q  - 6 q  + 3 q  - q


     q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{Knot[9, 39], Knot[11, NonAlternating, 11], 
  Knot[11, NonAlternating, 112]}
In[12]:=
A2Invariant[Knot[9, 39]][q]
Out[12]=  
      -4    -2      2    4      6    8    10    12      14      16

-1 + q   - q   + 3 q  - q  + 2 q  + q  - q   + q   - 2 q   + 2 q   - 



  20      22    24    26


  q   + 2 q   - q   - q
In[13]:=
Kauffman[Knot[9, 39]][a, z]
Out[13]=  
                                                   2      2       2

 -8   2    2    2    z    z    3 z   z     2   3 z    9 z    12 z



-a   - -- - -- - -- + -- - -- - --- - -- - z  + ---- + ---- + ----- + 



       6    4    2    9    7    5     3          8      6      4
      a    a    a    a    a    a     a          a      a      a

    2      3      3       3      3      3           4       4
 5 z    2 z    2 z    12 z    5 z    3 z     4   6 z    13 z
 ---- - ---- + ---- + ----- + ---- - ---- + z  - ---- - ----- - 
   2      9      7      5       3     a            8      6
  a      a      a      a       a                  a      a

     4      4    5      5       5      5      5      6      6      6
 15 z    7 z    z    7 z    18 z    7 z    3 z    3 z    3 z    5 z
 ----- - ---- + -- - ---- - ----- - ---- + ---- + ---- + ---- + ---- + 
   4       2     9     7      5       3     a       8      6      4
  a       a     a     a      a       a             a      a      a

    6      7      7      7      8      8
 5 z    4 z    9 z    5 z    2 z    2 z
 ---- + ---- + ---- + ---- + ---- + ----
   2      7      5      3      6      4


   a      a      a      a      a      a
In[14]:=
{Vassiliev[2][Knot[9, 39]], Vassiliev[3][Knot[9, 39]]}
Out[14]=  
{0, 4}
In[15]:=
Kh[Knot[9, 39]][q, t]
Out[15]=  
         3     1      2    q      3        5        5  2      7  2

4 q + 3 q  + ----- + --- + - + 5 q  t + 3 q  t + 5 q  t  + 5 q  t  + 



             3  2   q t   t
            q  t

    7  3      9  3      9  4      11  4      11  5      13  5
 4 q  t  + 5 q  t  + 4 q  t  + 4 q   t  + 2 q   t  + 4 q   t  + 

  13  6      15  6    17  7


  q   t  + 2 q   t  + q   t
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