10 34

10_33

10_35

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10 34 Quick Notes

10 34 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3849 X13,17,14,16 X5,15,6,14 X15,7,16,6 X9,1,10,20 X11,19,12,18 X17,13,18,12 X19,11,20,10 X7283
Gauss code	-1, 10, -2, 1, -4, 5, -10, 2, -6, 9, -7, 8, -3, 4, -5, 3, -8, 7, -9, 6
Dowker-Thistlethwaite code	4 8 14 2 20 18 16 6 12 10
Conway Notation	[2512]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 2 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-4][-8] Hyperbolic Volume 8.42227 A-Polynomial See Data:10 34/A-polynomial

[edit Notes for 10 34's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $2$ Rasmussen s-Invariant 0

[edit Notes for 10 34's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$3t^{2}-9t+13-9t^{-1}+3t^{-2}$
Conway polynomial	$3z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 37, 0 }
Jones polynomial	$-q^{7}+2q^{6}-3q^{5}+4q^{4}-5q^{3}+6q^{2}-5q+5-3q^{-1}+2q^{-2}-q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{-2}+z^{4}a^{-4}+z^{4}-a^{2}z^{2}+z^{2}a^{-2}+2z^{2}a^{-4}-z^{2}a^{-6}+2z^{2}-a^{2}+a^{-4}-a^{-6}+2$
Kauffman polynomial (db, data sources)	$z^{9}a^{-3}+z^{9}a^{-5}+2z^{8}a^{-2}+4z^{8}a^{-4}+2z^{8}a^{-6}+2z^{7}a^{-1}-z^{7}a^{-3}-2z^{7}a^{-5}+z^{7}a^{-7}-5z^{6}a^{-2}-17z^{6}a^{-4}-10z^{6}a^{-6}+2z^{6}+2az^{5}-2z^{5}a^{-1}-5z^{5}a^{-3}-6z^{5}a^{-5}-5z^{5}a^{-7}+2a^{2}z^{4}+4z^{4}a^{-2}+20z^{4}a^{-4}+14z^{4}a^{-6}+a^{3}z^{3}-z^{3}a^{-1}+5z^{3}a^{-3}+12z^{3}a^{-5}+7z^{3}a^{-7}-2a^{2}z^{2}-3z^{2}a^{-2}-8z^{2}a^{-4}-6z^{2}a^{-6}-3z^{2}-a^{3}z-az-za^{-3}-4za^{-5}-3za^{-7}+a^{2}+a^{-4}+a^{-6}+2$
The A2 invariant	$-q^{10}-q^{4}+2q^{2}+1+q^{-2}+q^{-4}+q^{-8}+q^{-14}-q^{-16}-q^{-22}$
The G2 invariant	$q^{52}-q^{50}+2q^{48}-2q^{46}-3q^{40}+4q^{38}-5q^{36}+4q^{34}-4q^{32}+3q^{28}-6q^{26}+7q^{24}-7q^{22}+4q^{20}-2q^{18}-2q^{16}+5q^{14}-5q^{12}+6q^{10}-2q^{8}+2q^{6}+q^{2}+2-q^{-2}+5q^{-4}-4q^{-6}+4q^{-8}-q^{-10}-2q^{-12}+4q^{-14}-3q^{-16}+2q^{-18}+2q^{-20}-4q^{-22}+3q^{-24}+q^{-26}-6q^{-28}+11q^{-30}-11q^{-32}+7q^{-34}+q^{-36}-7q^{-38}+15q^{-40}-15q^{-42}+12q^{-44}-6q^{-46}-2q^{-48}+8q^{-50}-10q^{-52}+11q^{-54}-6q^{-56}+2q^{-58}+3q^{-60}-6q^{-62}+5q^{-64}-q^{-66}-6q^{-68}+9q^{-70}-9q^{-72}+4q^{-74}+4q^{-76}-12q^{-78}+16q^{-80}-15q^{-82}+7q^{-84}-10q^{-88}+12q^{-90}-12q^{-92}+8q^{-94}-2q^{-96}-2q^{-98}+3q^{-100}-4q^{-102}+3q^{-104}-q^{-106}+q^{-108}$

Further Quantum Invariants

Further quantum knot invariants for 10_34.

A1 Invariants.

Weight	Invariant
1	$-q^{7}+q^{5}-q^{3}+2q+q^{-3}+q^{-5}-q^{-7}+q^{-9}-q^{-11}+q^{-13}-q^{-15}$
2	$q^{20}-q^{18}-q^{16}+2q^{14}-2q^{12}-q^{10}+2q^{8}-2q^{6}+q^{4}+2q^{2}-2+2q^{-2}+3q^{-4}-q^{-6}-q^{-8}+3q^{-10}+q^{-12}-2q^{-14}+2q^{-18}-2q^{-20}-2q^{-22}+3q^{-24}-q^{-26}-4q^{-28}+3q^{-30}+2q^{-32}-4q^{-34}+q^{-36}+3q^{-38}-2q^{-40}-q^{-42}+q^{-44}$
3	$-q^{39}+q^{37}+q^{35}-2q^{31}+q^{29}+3q^{27}-q^{25}-4q^{23}-2q^{21}+5q^{19}+2q^{17}-4q^{15}-6q^{13}+3q^{11}+8q^{9}+q^{7}-11q^{5}-2q^{3}+8q+8q^{-1}-5q^{-3}-7q^{-5}+2q^{-7}+7q^{-9}+6q^{-11}-2q^{-13}-6q^{-15}-q^{-17}+9q^{-19}+3q^{-21}-8q^{-23}-6q^{-25}+7q^{-27}+3q^{-29}-7q^{-31}-5q^{-33}+6q^{-35}+5q^{-37}-3q^{-39}-6q^{-41}+q^{-43}+5q^{-45}+2q^{-47}-4q^{-49}-6q^{-51}+q^{-53}+8q^{-55}+3q^{-57}-6q^{-59}-7q^{-61}+4q^{-63}+9q^{-65}-q^{-67}-8q^{-69}-3q^{-71}+6q^{-73}+5q^{-75}-3q^{-77}-4q^{-79}+2q^{-83}+q^{-85}-q^{-87}$
4	$q^{64}-q^{62}-q^{60}+3q^{54}-3q^{52}-q^{50}+q^{48}+2q^{46}+7q^{44}-7q^{42}-6q^{40}-q^{38}+7q^{36}+15q^{34}-10q^{32}-16q^{30}-8q^{28}+14q^{26}+29q^{24}-9q^{22}-29q^{20}-21q^{18}+19q^{16}+46q^{14}-39q^{10}-39q^{8}+15q^{6}+57q^{4}+19q^{2}-30-48q^{-2}-7q^{-4}+43q^{-6}+36q^{-8}-33q^{-12}-28q^{-14}+6q^{-16}+23q^{-18}+25q^{-20}+3q^{-22}-25q^{-24}-22q^{-26}+q^{-28}+25q^{-30}+21q^{-32}-14q^{-34}-27q^{-36}-7q^{-38}+21q^{-40}+24q^{-42}-13q^{-44}-32q^{-46}-6q^{-48}+25q^{-50}+29q^{-52}-10q^{-54}-38q^{-56}-13q^{-58}+22q^{-60}+35q^{-62}+3q^{-64}-34q^{-66}-22q^{-68}+6q^{-70}+29q^{-72}+19q^{-74}-14q^{-76}-18q^{-78}-12q^{-80}+7q^{-82}+17q^{-84}+4q^{-86}+3q^{-88}-11q^{-90}-13q^{-92}-q^{-94}+2q^{-96}+19q^{-98}+8q^{-100}-7q^{-102}-13q^{-104}-18q^{-106}+10q^{-108}+17q^{-110}+10q^{-112}-2q^{-114}-21q^{-116}-7q^{-118}+4q^{-120}+12q^{-122}+11q^{-124}-7q^{-126}-7q^{-128}-5q^{-130}+q^{-132}+6q^{-134}+q^{-136}-2q^{-140}-q^{-142}+q^{-144}$
5	$-q^{95}+q^{93}+q^{91}-q^{85}-q^{83}+q^{81}+q^{79}-2q^{77}-q^{75}-2q^{73}+2q^{71}+6q^{69}+3q^{67}-5q^{65}-9q^{63}-5q^{61}+6q^{59}+16q^{57}+12q^{55}-9q^{53}-25q^{51}-16q^{49}+13q^{47}+30q^{45}+22q^{43}-12q^{41}-44q^{39}-26q^{37}+24q^{35}+52q^{33}+23q^{31}-34q^{29}-68q^{27}-28q^{25}+56q^{23}+86q^{21}+27q^{19}-73q^{17}-115q^{15}-36q^{13}+87q^{11}+139q^{9}+59q^{7}-81q^{5}-162q^{3}-91q+67q^{-1}+160q^{-3}+122q^{-5}-14q^{-7}-138q^{-9}-140q^{-11}-30q^{-13}+93q^{-15}+132q^{-17}+75q^{-19}-28q^{-21}-102q^{-23}-99q^{-25}-25q^{-27}+60q^{-29}+93q^{-31}+66q^{-33}-15q^{-35}-85q^{-37}-80q^{-39}-9q^{-41}+65q^{-43}+81q^{-45}+18q^{-47}-60q^{-49}-76q^{-51}-14q^{-53}+61q^{-55}+75q^{-57}+8q^{-59}-72q^{-61}-83q^{-63}-10q^{-65}+78q^{-67}+100q^{-69}+19q^{-71}-84q^{-73}-115q^{-75}-42q^{-77}+76q^{-79}+134q^{-81}+71q^{-83}-60q^{-85}-138q^{-87}-98q^{-89}+26q^{-91}+134q^{-93}+128q^{-95}+9q^{-97}-114q^{-99}-138q^{-101}-50q^{-103}+77q^{-105}+135q^{-107}+81q^{-109}-33q^{-111}-114q^{-113}-95q^{-115}-9q^{-117}+73q^{-119}+91q^{-121}+42q^{-123}-30q^{-125}-67q^{-127}-49q^{-129}-6q^{-131}+29q^{-133}+37q^{-135}+25q^{-137}+3q^{-139}-14q^{-141}-18q^{-143}-21q^{-145}-15q^{-147}-q^{-149}+19q^{-151}+29q^{-153}+22q^{-155}+3q^{-157}-25q^{-159}-37q^{-161}-23q^{-163}+9q^{-165}+32q^{-167}+33q^{-169}+14q^{-171}-17q^{-173}-33q^{-175}-25q^{-177}+19q^{-181}+24q^{-183}+15q^{-185}-6q^{-187}-17q^{-189}-14q^{-191}-3q^{-193}+5q^{-195}+9q^{-197}+7q^{-199}-q^{-201}-4q^{-203}-3q^{-205}-q^{-207}+2q^{-211}+q^{-213}-q^{-215}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{10}-q^{4}+2q^{2}+1+q^{-2}+q^{-4}+q^{-8}+q^{-14}-q^{-16}-q^{-22}$
1,1	$q^{28}-2q^{26}+4q^{24}-6q^{22}+9q^{20}-12q^{18}+14q^{16}-18q^{14}+17q^{12}-20q^{10}+16q^{8}-16q^{6}+12q^{4}-8q^{2}+8-2q^{-2}+10q^{-4}-4q^{-6}+10q^{-8}-2q^{-10}+q^{-12}+12q^{-14}-24q^{-16}+38q^{-18}-53q^{-20}+62q^{-22}-70q^{-24}+68q^{-26}-65q^{-28}+52q^{-30}-38q^{-32}+18q^{-34}+q^{-36}-20q^{-38}+36q^{-40}-46q^{-42}+49q^{-44}-46q^{-46}+40q^{-48}-30q^{-50}+19q^{-52}-12q^{-54}+6q^{-56}-2q^{-58}+q^{-60}$
2,0	$q^{26}-q^{22}+q^{18}-q^{16}-2q^{14}+q^{12}+q^{10}-4q^{8}-3q^{6}+3q^{4}-1+2q^{-2}+5q^{-4}+3q^{-6}+q^{-8}+3q^{-10}+2q^{-12}+2q^{-16}-2q^{-20}-q^{-22}+q^{-24}-q^{-26}-2q^{-28}-q^{-30}+q^{-32}-3q^{-36}-q^{-38}+q^{-40}+q^{-42}-q^{-44}-q^{-46}+2q^{-48}+q^{-50}-q^{-52}-q^{-54}+q^{-58}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{22}-q^{20}+2q^{16}-3q^{14}-3q^{12}+2q^{10}-2q^{8}-3q^{6}+4q^{4}+2q^{2}+3q^{-2}+3q^{-4}-q^{-6}+q^{-8}+2q^{-10}+2q^{-12}-q^{-14}+q^{-16}+3q^{-18}-2q^{-20}+2q^{-24}-2q^{-26}-q^{-28}-3q^{-32}-q^{-34}+q^{-36}-2q^{-38}+q^{-40}+q^{-42}-q^{-44}+q^{-46}$
1,0,0	$-q^{13}-q^{9}-q^{5}+2q^{3}+q+2q^{-1}+q^{-3}+q^{-5}+q^{-11}+q^{-15}+q^{-19}-q^{-21}-q^{-25}-q^{-29}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{22}+q^{20}-2q^{18}+2q^{16}-3q^{14}+3q^{12}-4q^{10}+4q^{8}-3q^{6}+4q^{4}-2q^{2}+2+q^{-2}-q^{-4}+5q^{-6}-5q^{-8}+8q^{-10}-8q^{-12}+9q^{-14}-9q^{-16}+7q^{-18}-6q^{-20}+4q^{-22}-2q^{-24}+3q^{-28}-4q^{-30}+5q^{-32}-5q^{-34}+5q^{-36}-4q^{-38}+3q^{-40}-3q^{-42}+q^{-44}-q^{-46}$
1,0	$q^{36}-q^{32}-q^{30}+q^{28}+2q^{26}-3q^{22}-3q^{20}+3q^{16}+q^{14}-3q^{12}-3q^{10}+4q^{6}+2q^{4}-q^{2}-2+2q^{-2}+4q^{-4}+2q^{-6}-3q^{-8}-q^{-10}+3q^{-12}+4q^{-14}-q^{-16}-2q^{-18}+q^{-20}+3q^{-22}-2q^{-26}+q^{-28}+3q^{-30}-4q^{-34}-2q^{-36}+3q^{-38}+4q^{-40}-q^{-42}-6q^{-44}-2q^{-46}+4q^{-48}+3q^{-50}-3q^{-52}-5q^{-54}+4q^{-58}+q^{-60}-3q^{-62}-2q^{-64}+2q^{-66}+2q^{-68}-q^{-70}-q^{-72}+q^{-76}$

G2 Invariants.

Weight

Invariant

1,0

$q^{52}-q^{50}+2q^{48}-2q^{46}-3q^{40}+4q^{38}-5q^{36}+4q^{34}-4q^{32}+3q^{28}-6q^{26}+7q^{24}-7q^{22}+4q^{20}-2q^{18}-2q^{16}+5q^{14}-5q^{12}+6q^{10}-2q^{8}+2q^{6}+q^{2}+2-q^{-2}+5q^{-4}-4q^{-6}+4q^{-8}-q^{-10}-2q^{-12}+4q^{-14}-3q^{-16}+2q^{-18}+2q^{-20}-4q^{-22}+3q^{-24}+q^{-26}-6q^{-28}+11q^{-30}-11q^{-32}+7q^{-34}+q^{-36}-7q^{-38}+15q^{-40}-15q^{-42}+12q^{-44}-6q^{-46}-2q^{-48}+8q^{-50}-10q^{-52}+11q^{-54}-6q^{-56}+2q^{-58}+3q^{-60}-6q^{-62}+5q^{-64}-q^{-66}-6q^{-68}+9q^{-70}-9q^{-72}+4q^{-74}+4q^{-76}-12q^{-78}+16q^{-80}-15q^{-82}+7q^{-84}-10q^{-88}+12q^{-90}-12q^{-92}+8q^{-94}-2q^{-96}-2q^{-98}+3q^{-100}-4q^{-102}+3q^{-104}-q^{-106}+q^{-108}$

.

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["10 34"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$3t^{2}-9t+13-9t^{-1}+3t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$3z^{4}+3z^{2}+1$

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 37, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{7}+2q^{6}-3q^{5}+4q^{4}-5q^{3}+6q^{2}-5q+5-3q^{-1}+2q^{-2}-q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{4}a^{-2}+z^{4}a^{-4}+z^{4}-a^{2}z^{2}+z^{2}a^{-2}+2z^{2}a^{-4}-z^{2}a^{-6}+2z^{2}-a^{2}+a^{-4}-a^{-6}+2$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-3}+z^{9}a^{-5}+2z^{8}a^{-2}+4z^{8}a^{-4}+2z^{8}a^{-6}+2z^{7}a^{-1}-z^{7}a^{-3}-2z^{7}a^{-5}+z^{7}a^{-7}-5z^{6}a^{-2}-17z^{6}a^{-4}-10z^{6}a^{-6}+2z^{6}+2az^{5}-2z^{5}a^{-1}-5z^{5}a^{-3}-6z^{5}a^{-5}-5z^{5}a^{-7}+2a^{2}z^{4}+4z^{4}a^{-2}+20z^{4}a^{-4}+14z^{4}a^{-6}+a^{3}z^{3}-z^{3}a^{-1}+5z^{3}a^{-3}+12z^{3}a^{-5}+7z^{3}a^{-7}-2a^{2}z^{2}-3z^{2}a^{-2}-8z^{2}a^{-4}-6z^{2}a^{-6}-3z^{2}-a^{3}z-az-za^{-3}-4za^{-5}-3za^{-7}+a^{2}+a^{-4}+a^{-6}+2$

Vassiliev invariants

V2 and V3:

(3, 3)

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 $12$ $24$ $72$ $126$ $10$ $288$ $464$ $64$ $56$ $288$ $288$ $1512$ $120$ ${\frac {23311}{10}}$ ${\frac {2894}{15}}$ ${\frac {9782}{15}}$ ${\frac {209}{6}}$ ${\frac {271}{10}}$

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 $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s+1$, where $s=$0 is the signature of 10 34. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Computer Talk

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

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[10, 34]]
Out[2]=	10
In[3]:=	PD[Knot[10, 34]]
Out[3]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[13, 17, 14, 16], X[5, 15, 6, 14], X[15, 7, 16, 6], X[9, 1, 10, 20], X[11, 19, 12, 18], X[17, 13, 18, 12], X[19, 11, 20, 10], X[7, 2, 8, 3]]
In[4]:=	GaussCode[Knot[10, 34]]
Out[4]=	GaussCode[-1, 10, -2, 1, -4, 5, -10, 2, -6, 9, -7, 8, -3, 4, -5, 3, -8, 7, -9, 6]
In[5]:=	BR[Knot[10, 34]]
Out[5]=	BR[5, {1, 1, 1, 2, -1, 2, 3, -2, -4, 3, -4, -4}]
In[6]:=	alex = Alexander[Knot[10, 34]][t]
Out[6]=	3 9 2 13 + -- - - - 9 t + 3 t 2 t t
In[7]:=	Conway[Knot[10, 34]][z]
Out[7]=	2 4 1 + 3 z + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 34], Knot[10, 135]}
In[9]:=	{KnotDet[Knot[10, 34]], KnotSignature[Knot[10, 34]]}
Out[9]=	{37, 0}
In[10]:=	J=Jones[Knot[10, 34]][q]
Out[10]=	-3 2 3 2 3 4 5 6 7 5 - q + -- - - - 5 q + 6 q - 5 q + 4 q - 3 q + 2 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 34]}
In[12]:=	A2Invariant[Knot[10, 34]][q]
Out[12]=	-10 -4 2 2 4 8 14 16 22 1 - q - q + -- + q + q + q + q - q - q 2 q
In[13]:=	Kauffman[Knot[10, 34]][a, z]
Out[13]=	2 2 -6 -4 2 3 z 4 z z 3 2 6 z 8 z 2 + a + a + a - --- - --- - -- - a z - a z - 3 z - ---- - ---- - 7 5 3 6 4 a a a a a 2 3 3 3 3 4 4 3 z 2 2 7 z 12 z 5 z z 3 3 14 z 20 z ---- - 2 a z + ---- + ----- + ---- - -- + a z + ----- + ----- + 2 7 5 3 a 6 4 a a a a a a 4 5 5 5 5 6 4 z 2 4 5 z 6 z 5 z 2 z 5 6 10 z ---- + 2 a z - ---- - ---- - ---- - ---- + 2 a z + 2 z - ----- - 2 7 5 3 a 6 a a a a a 6 6 7 7 7 7 8 8 8 9 9 17 z 5 z z 2 z z 2 z 2 z 4 z 2 z z z ----- - ---- + -- - ---- - -- + ---- + ---- + ---- + ---- + -- + -- 4 2 7 5 3 a 6 4 2 5 3 a a a a a a a a a a
In[14]:=	{Vassiliev[2][Knot[10, 34]], Vassiliev[3][Knot[10, 34]]}
Out[14]=	{0, 3}
In[15]:=	Kh[Knot[10, 34]][q, t]
Out[15]=	3 1 1 1 2 1 3 - + 3 q + ----- + ----- + ----- + ---- + --- + 3 q t + 2 q t + q 7 3 5 2 3 2 3 q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 9 5 3 q t + 3 q t + 2 q t + 3 q t + 2 q t + 2 q t + q t + 11 5 11 6 13 6 15 7 2 q t + q t + q t + q t