9 5

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9 4.gif

9_4

9 6.gif

9_6

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Visit 9 5's page at Knotilus!

Visit 9 5's page at the original Knot Atlas!

9 5 Quick Notes


9 5 Further Notes and Views

Knot presentations

Planar diagram presentation X6271 X14,6,15,5 X18,8,1,7 X16,10,17,9 X10,16,11,15 X8,18,9,17 X2,14,3,13 X12,4,13,3 X4,12,5,11
Gauss code 1, -7, 8, -9, 2, -1, 3, -6, 4, -5, 9, -8, 7, -2, 5, -4, 6, -3
Dowker-Thistlethwaite code 6 12 14 18 16 4 2 10 8
Conway Notation [513]

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 1
Bridge index 2
Super bridge index [math]\displaystyle{ \{4,6\} }[/math]
Nakanishi index 1
Maximal Thurston-Bennequin number [1][-12]
Hyperbolic Volume 5.69844
A-Polynomial See Data:9 5/A-polynomial

[edit Notes for 9 5'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{ 1 }[/math]
Rasmussen s-Invariant -2

[edit Notes for 9 5's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ 6 t-11+6 t^{-1} }[/math]
Conway polynomial [math]\displaystyle{ 6 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 23, 2 }
Jones polynomial [math]\displaystyle{ -q^{10}+q^9-2 q^8+3 q^7-3 q^6+4 q^5-3 q^4+3 q^3-2 q^2+q }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^2 a^{-2} +2 z^2 a^{-4} +2 z^2 a^{-6} +z^2 a^{-8} + a^{-4} + a^{-6} - a^{-10} }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^8 a^{-8} +z^8 a^{-10} +2 z^7 a^{-7} +3 z^7 a^{-9} +z^7 a^{-11} +3 z^6 a^{-6} -2 z^6 a^{-8} -5 z^6 a^{-10} +3 z^5 a^{-5} -5 z^5 a^{-7} -14 z^5 a^{-9} -6 z^5 a^{-11} +3 z^4 a^{-4} -7 z^4 a^{-6} -3 z^4 a^{-8} +7 z^4 a^{-10} +2 z^3 a^{-3} -4 z^3 a^{-5} +z^3 a^{-7} +18 z^3 a^{-9} +11 z^3 a^{-11} +z^2 a^{-2} -3 z^2 a^{-4} +3 z^2 a^{-6} +4 z^2 a^{-8} -3 z^2 a^{-10} -6 z a^{-9} -6 z a^{-11} + a^{-4} - a^{-6} + a^{-10} }[/math]
The A2 invariant [math]\displaystyle{ q^{-2} - q^{-4} + q^{-8} + q^{-12} + q^{-14} + q^{-16} + q^{-18} + q^{-22} - q^{-26} - q^{-30} - q^{-32} }[/math]
The G2 invariant [math]\displaystyle{ q^{-10} - q^{-12} + q^{-14} - q^{-16} - q^{-22} +3 q^{-24} -2 q^{-26} +2 q^{-28} - q^{-30} + q^{-34} - q^{-36} +3 q^{-38} -2 q^{-40} + q^{-42} +2 q^{-48} - q^{-50} + q^{-52} - q^{-54} + q^{-56} - q^{-60} + q^{-62} + q^{-66} + q^{-72} + q^{-74} +2 q^{-78} -2 q^{-80} +5 q^{-82} - q^{-84} - q^{-86} +5 q^{-88} -4 q^{-90} +6 q^{-92} -2 q^{-94} - q^{-96} +3 q^{-98} -2 q^{-100} +4 q^{-102} -3 q^{-104} - q^{-110} + q^{-112} -2 q^{-114} - q^{-116} + q^{-118} -3 q^{-120} -2 q^{-124} -2 q^{-126} +3 q^{-128} -6 q^{-130} +3 q^{-132} -2 q^{-134} -2 q^{-136} +4 q^{-138} -5 q^{-140} +3 q^{-142} - q^{-144} + q^{-148} -2 q^{-150} +2 q^{-152} + q^{-156} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 9_5.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^{-1} - q^{-3} + q^{-5} + q^{-9} + q^{-11} + q^{-15} - q^{-17} - q^{-21} }[/math]
2 [math]\displaystyle{ q^{-2} - q^{-4} +2 q^{-8} - q^{-10} + q^{-12} + q^{-14} - q^{-16} + q^{-18} + q^{-20} +2 q^{-26} - q^{-30} + q^{-34} - q^{-36} - q^{-38} + q^{-40} - q^{-42} -2 q^{-44} + q^{-46} -2 q^{-50} + q^{-52} + q^{-54} - q^{-56} + q^{-60} }[/math]
3 [math]\displaystyle{ q^{-3} - q^{-5} + q^{-9} + q^{-11} - q^{-15} + q^{-17} + q^{-19} + q^{-21} - q^{-25} + q^{-27} +2 q^{-29} + q^{-31} -3 q^{-33} +2 q^{-37} +2 q^{-39} - q^{-43} - q^{-45} + q^{-47} +3 q^{-49} + q^{-51} -3 q^{-53} -2 q^{-55} +2 q^{-57} - q^{-59} -3 q^{-61} -2 q^{-63} + q^{-65} - q^{-71} + q^{-73} + q^{-75} -2 q^{-79} - q^{-81} + q^{-83} +2 q^{-85} - q^{-87} -2 q^{-89} +3 q^{-93} +2 q^{-95} -2 q^{-97} -2 q^{-99} + q^{-101} +3 q^{-103} -2 q^{-107} - q^{-109} + q^{-111} + q^{-113} - q^{-117} }[/math]
4 [math]\displaystyle{ q^{-4} - q^{-6} + q^{-10} +2 q^{-14} -2 q^{-16} + q^{-20} + q^{-22} +4 q^{-24} -4 q^{-26} -2 q^{-28} + q^{-30} +4 q^{-32} +5 q^{-34} -5 q^{-36} -5 q^{-38} +7 q^{-42} +8 q^{-44} -4 q^{-46} -8 q^{-48} -3 q^{-50} +6 q^{-52} +10 q^{-54} -7 q^{-58} -8 q^{-60} - q^{-62} +7 q^{-64} +4 q^{-66} +2 q^{-68} -3 q^{-70} -6 q^{-72} - q^{-74} +4 q^{-76} +6 q^{-78} +2 q^{-80} -6 q^{-82} -6 q^{-84} - q^{-86} +4 q^{-88} +3 q^{-90} -4 q^{-92} -6 q^{-94} +3 q^{-98} + q^{-100} -4 q^{-102} -5 q^{-104} +2 q^{-106} +5 q^{-108} +3 q^{-110} -3 q^{-112} -5 q^{-114} +2 q^{-116} +5 q^{-118} +5 q^{-120} + q^{-122} -5 q^{-124} -2 q^{-126} - q^{-128} +3 q^{-130} +4 q^{-132} -2 q^{-134} -2 q^{-136} -4 q^{-138} - q^{-140} +5 q^{-142} +3 q^{-144} +3 q^{-146} -3 q^{-148} -5 q^{-150} - q^{-152} + q^{-154} +6 q^{-156} +2 q^{-158} -3 q^{-160} -4 q^{-162} -4 q^{-164} +3 q^{-166} +4 q^{-168} +2 q^{-170} - q^{-172} -5 q^{-174} - q^{-176} + q^{-178} +2 q^{-180} +2 q^{-182} - q^{-184} - q^{-186} - q^{-188} + q^{-192} }[/math]
5 [math]\displaystyle{ q^{-5} - q^{-7} + q^{-11} + q^{-15} - q^{-19} +2 q^{-23} +2 q^{-25} -2 q^{-29} -3 q^{-31} + q^{-33} +5 q^{-35} +6 q^{-37} -2 q^{-39} -7 q^{-41} -5 q^{-43} +3 q^{-45} +11 q^{-47} +8 q^{-49} -4 q^{-51} -13 q^{-53} -7 q^{-55} +7 q^{-57} +14 q^{-59} +8 q^{-61} -7 q^{-63} -17 q^{-65} -11 q^{-67} +8 q^{-69} +21 q^{-71} +13 q^{-73} -4 q^{-75} -17 q^{-77} -18 q^{-79} - q^{-81} +17 q^{-83} +17 q^{-85} +5 q^{-87} -8 q^{-89} -17 q^{-91} -12 q^{-93} - q^{-95} +10 q^{-97} +14 q^{-99} +8 q^{-101} -2 q^{-103} -13 q^{-105} -15 q^{-107} -6 q^{-109} +8 q^{-111} +15 q^{-113} +10 q^{-115} -3 q^{-117} -13 q^{-119} -11 q^{-121} + q^{-123} +8 q^{-125} +7 q^{-127} -2 q^{-129} -8 q^{-131} -5 q^{-133} +2 q^{-135} +5 q^{-137} +4 q^{-139} -6 q^{-141} -10 q^{-143} -3 q^{-145} +7 q^{-147} +12 q^{-149} +8 q^{-151} -6 q^{-153} -14 q^{-155} -8 q^{-157} +5 q^{-159} +15 q^{-161} +14 q^{-163} + q^{-165} -13 q^{-167} -14 q^{-169} -3 q^{-171} +10 q^{-173} +17 q^{-175} +8 q^{-177} -6 q^{-179} -14 q^{-181} -12 q^{-183} + q^{-185} +10 q^{-187} +11 q^{-189} +4 q^{-191} -5 q^{-193} -11 q^{-195} -10 q^{-197} - q^{-199} +6 q^{-201} +8 q^{-203} +7 q^{-205} + q^{-207} -6 q^{-209} -8 q^{-211} -4 q^{-213} +6 q^{-217} +8 q^{-219} +5 q^{-221} -2 q^{-223} -7 q^{-225} -8 q^{-227} -5 q^{-229} +2 q^{-231} +8 q^{-233} +8 q^{-235} +3 q^{-237} -4 q^{-239} -8 q^{-241} -7 q^{-243} - q^{-245} +5 q^{-247} +8 q^{-249} +5 q^{-251} - q^{-253} -5 q^{-255} -6 q^{-257} -3 q^{-259} +2 q^{-261} +5 q^{-263} +3 q^{-265} + q^{-267} - q^{-269} -3 q^{-271} -2 q^{-273} + q^{-277} + q^{-279} + q^{-281} - q^{-285} }[/math]

A2 Invariants.

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

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{-4} - q^{-6} - q^{-8} +2 q^{-10} - q^{-14} +2 q^{-16} + q^{-18} +2 q^{-22} +2 q^{-24} + q^{-28} + q^{-30} + q^{-32} +2 q^{-36} +3 q^{-38} + q^{-40} + q^{-44} -3 q^{-46} -3 q^{-48} -3 q^{-50} -4 q^{-52} -2 q^{-54} + q^{-60} +2 q^{-62} + q^{-66} }[/math]
1,0,0 [math]\displaystyle{ q^{-3} - q^{-5} + q^{-11} + q^{-15} + q^{-17} + q^{-19} + q^{-21} + q^{-23} + q^{-25} + q^{-29} - q^{-35} - q^{-39} - q^{-41} - q^{-43} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{-4} - q^{-6} + q^{-8} -2 q^{-10} +2 q^{-12} - q^{-14} +2 q^{-16} - q^{-18} +2 q^{-20} +2 q^{-26} - q^{-28} +3 q^{-30} -3 q^{-32} +4 q^{-34} -4 q^{-36} +3 q^{-38} -3 q^{-40} +2 q^{-42} - q^{-44} + q^{-46} + q^{-48} - q^{-50} +2 q^{-52} -2 q^{-54} +2 q^{-56} -2 q^{-58} + q^{-60} -2 q^{-62} - q^{-66} }[/math]
1,0 [math]\displaystyle{ q^{-6} - q^{-10} - q^{-12} +2 q^{-16} + q^{-18} - q^{-20} - q^{-22} +2 q^{-26} + q^{-28} - q^{-32} + q^{-34} +2 q^{-36} + q^{-38} - q^{-40} + q^{-44} +2 q^{-46} - q^{-50} +2 q^{-54} + q^{-56} + q^{-60} +2 q^{-62} + q^{-64} - q^{-66} - q^{-68} + q^{-70} + q^{-72} - q^{-74} -4 q^{-76} -2 q^{-78} -2 q^{-84} -3 q^{-86} - q^{-88} + q^{-90} + q^{-92} - q^{-94} - q^{-96} + q^{-98} +2 q^{-100} + q^{-108} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{-10} - q^{-12} + q^{-14} - q^{-16} - q^{-22} +3 q^{-24} -2 q^{-26} +2 q^{-28} - q^{-30} + q^{-34} - q^{-36} +3 q^{-38} -2 q^{-40} + q^{-42} +2 q^{-48} - q^{-50} + q^{-52} - q^{-54} + q^{-56} - q^{-60} + q^{-62} + q^{-66} + q^{-72} + q^{-74} +2 q^{-78} -2 q^{-80} +5 q^{-82} - q^{-84} - q^{-86} +5 q^{-88} -4 q^{-90} +6 q^{-92} -2 q^{-94} - q^{-96} +3 q^{-98} -2 q^{-100} +4 q^{-102} -3 q^{-104} - q^{-110} + q^{-112} -2 q^{-114} - q^{-116} + q^{-118} -3 q^{-120} -2 q^{-124} -2 q^{-126} +3 q^{-128} -6 q^{-130} +3 q^{-132} -2 q^{-134} -2 q^{-136} +4 q^{-138} -5 q^{-140} +3 q^{-142} - q^{-144} + q^{-148} -2 q^{-150} +2 q^{-152} + q^{-156} }[/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 5"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 6 t-11+6 t^{-1} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 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]= { 23, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^{10}+q^9-2 q^8+3 q^7-3 q^6+4 q^5-3 q^4+3 q^3-2 q^2+q }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^2 a^{-2} +2 z^2 a^{-4} +2 z^2 a^{-6} +z^2 a^{-8} + a^{-4} + a^{-6} - a^{-10} }[/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} +2 z^7 a^{-7} +3 z^7 a^{-9} +z^7 a^{-11} +3 z^6 a^{-6} -2 z^6 a^{-8} -5 z^6 a^{-10} +3 z^5 a^{-5} -5 z^5 a^{-7} -14 z^5 a^{-9} -6 z^5 a^{-11} +3 z^4 a^{-4} -7 z^4 a^{-6} -3 z^4 a^{-8} +7 z^4 a^{-10} +2 z^3 a^{-3} -4 z^3 a^{-5} +z^3 a^{-7} +18 z^3 a^{-9} +11 z^3 a^{-11} +z^2 a^{-2} -3 z^2 a^{-4} +3 z^2 a^{-6} +4 z^2 a^{-8} -3 z^2 a^{-10} -6 z a^{-9} -6 z a^{-11} + a^{-4} - a^{-6} + a^{-10} }[/math]

Vassiliev invariants

V2 and V3: (6, 15)
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{ 120 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 780 }[/math] [math]\displaystyle{ 140 }[/math] [math]\displaystyle{ 2880 }[/math] [math]\displaystyle{ 5520 }[/math] [math]\displaystyle{ 960 }[/math] [math]\displaystyle{ 920 }[/math] [math]\displaystyle{ 2304 }[/math] [math]\displaystyle{ 7200 }[/math] [math]\displaystyle{ 18720 }[/math] [math]\displaystyle{ 3360 }[/math] [math]\displaystyle{ \frac{196391}{5} }[/math] [math]\displaystyle{ -\frac{14492}{15} }[/math] [math]\displaystyle{ \frac{274084}{15} }[/math] [math]\displaystyle{ \frac{1193}{3} }[/math] [math]\displaystyle{ \frac{13031}{5} }[/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 5. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
 0123456789χ
21         1-1
19          0
17       21 -1
15      1   1
13     22   0
11    21    1
9   12     1
7  22      0
5  1       1
312        -1
11         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, 5]]
Out[2]=  
9
In[3]:=
PD[Knot[9, 5]]
Out[3]=  
PD[X[6, 2, 7, 1], X[14, 6, 15, 5], X[18, 8, 1, 7], X[16, 10, 17, 9], 

 X[10, 16, 11, 15], X[8, 18, 9, 17], X[2, 14, 3, 13], X[12, 4, 13, 3], 



  X[4, 12, 5, 11]]
In[4]:=
GaussCode[Knot[9, 5]]
Out[4]=  
GaussCode[1, -7, 8, -9, 2, -1, 3, -6, 4, -5, 9, -8, 7, -2, 5, -4, 6, -3]
In[5]:=
BR[Knot[9, 5]]
Out[5]=  
BR[5, {1, 1, 2, -1, 2, 2, 3, -2, 3, 4, -3, 4}]
In[6]:=
alex = Alexander[Knot[9, 5]][t]
Out[6]=  
      6

-11 + - + 6 t


      t
In[7]:=
Conway[Knot[9, 5]][z]
Out[7]=  
       2
1 + 6 z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[9, 5]}
In[9]:=
{KnotDet[Knot[9, 5]], KnotSignature[Knot[9, 5]]}
Out[9]=  
{23, 2}
In[10]:=
J=Jones[Knot[9, 5]][q]
Out[10]=  
       2      3      4      5      6      7      8    9    10
q - 2 q  + 3 q  - 3 q  + 4 q  - 3 q  + 3 q  - 2 q  + q  - q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{Knot[9, 5]}
In[12]:=
A2Invariant[Knot[9, 5]][q]
Out[12]=  
 2    4    8    12    14    16    18    22    26    30    32
q  - q  + q  + q   + q   + q   + q   + q   - q   - q   - q
In[13]:=
Kauffman[Knot[9, 5]][a, z]
Out[13]=  
                                  2      2      2      2    2       3

-10    -6    -4   6 z   6 z   3 z    4 z    3 z    3 z    z    11 z



a    - a   + a   - --- - --- - ---- + ---- + ---- - ---- + -- + ----- + 



                   11    9     10      8      6      4     2     11
                  a     a     a       a      a      a     a     a

     3    3      3      3      4      4      4      4      5       5
 18 z    z    4 z    2 z    7 z    3 z    7 z    3 z    6 z    14 z
 ----- + -- - ---- + ---- + ---- - ---- - ---- + ---- - ---- - ----- - 
   9      7     5      3     10      8      6      4     11      9
  a      a     a      a     a       a      a      a     a       a

    5      5      6      6      6    7       7      7    8     8
 5 z    3 z    5 z    2 z    3 z    z     3 z    2 z    z     z
 ---- + ---- - ---- - ---- + ---- + --- + ---- + ---- + --- + --
   7      5     10      8      6     11     9      7     10    8


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

q + q  + 2 q  t + q  t  + 2 q  t  + 2 q  t  + q  t  + 2 q  t  + 



    11  4    11  5      13  5      13  6    15  6      17  7
 2 q   t  + q   t  + 2 q   t  + 2 q   t  + q   t  + 2 q   t  + 

  17  8    21  9


  q   t  + q   t
Retrieved from "https://katlas.org/index.php?title=9_5&oldid=25942"