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L11n340 L11n342

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X18,12,19,11 X14,8,15,7 X8,14,9,13 X19,22,20,13 X15,20,16,21 X21,16,22,17 X12,18,5,17 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

- 2q-6 + 5q-5 - 8q-4 + 11q-3 - 10q-2 + 11q-1 - 8 + 6q - 2q2 + q3

A2 (sl(3)) Invariant:

- q-20 - 3q-18 - 3q-14 + 4q-10 + 3q-8 + 9q-6 + 4q-4 + 6q-2 + 3 + 3q4 + q8 + q10

HOMFLY-PT Polynomial:

2a-2 + a-2z2 + 2z-2 - 4z2 - 2z4 - 5a2z-2 - 7a2 - 2a2z2 + 2a2z4 + a2z6 + 4a4z-2 + 6a4 + a4z2 - a4z4 - a6z-2 - a6

Kauffman Polynomial:

- 2a-2 + 5a-2z2 - 4a-2z4 + a-2z6 + a-1z + 2a-1z3 - 5a-1z5 + 2a-1z7 - 2z-2 + 7 - 9z2 + 8z4 - 8z6 + 3z8 + 5az-1 - 18az + 28az3 - 25az5 + 6az7 + az9 - 5a2z-2 + 21a2 - 34a2z2 + 27a2z4 - 19a2z6 + 7a2z8 + 9a3z-1 - 33a3z + 43a3z3 - 29a3z5 + 8a3z7 + a3z9 - 4a4z-2 + 16a4 - 24a4z2 + 19a4z4 - 9a4z6 + 4a4z8 + 5a5z-1 - 17a5z + 20a5z3 - 9a5z5 + 4a5z7 - a6z-2 + 3a6 - 4a6z2 + 4a6z4 + a6z6 + a7z-1 - 3a7z + 3a7z3

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                   1 j = 5                 2 1 j = 3               4     j = 1             4 2     j = -1           7 4       j = -3         5 6         j = -5       6 5           j = -7     2 5             j = -9   3 6               j = -11   3                 j = -13 2

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 341]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 341]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[18, 12, 19, 11], X[14, 8, 15, 7], > X[8, 14, 9, 13], X[19, 22, 20, 13], X[15, 20, 16, 21], X[21, 16, 22, 17], > X[12, 18, 5, 17], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, -5, 11, -2, 3, -9}, > {5, -4, -7, 8, 9, -3, -6, 7, -8, 6}]
In[6]:=	Jones[L][q]
Out[6]=	2 5 8 11 10 11 2 3 -8 - -- + -- - -- + -- - -- + -- + 6 q - 2 q + q 6 5 4 3 2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 3 3 4 3 9 4 6 4 8 10 3 - q - --- - --- + --- + -- + -- + -- + -- + 3 q + q + q 18 14 10 8 6 4 2 q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 341]][a, z]
Out[8]=	2 4 6 2 2 2 4 6 2 5 a 4 a a 2 z 2 2 4 2 -- - 7 a + 6 a - a + -- - ---- + ---- - -- - 4 z + -- - 2 a z + a z - 2 2 2 2 2 2 a z z z z a 4 2 4 4 4 2 6 > 2 z + 2 a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 341]][a, z]
Out[9]=	2 4 6 3 5 2 2 4 6 2 5 a 4 a a 5 a 9 a 5 a 7 - -- + 21 a + 16 a + 3 a - -- - ---- - ---- - -- + --- + ---- + ---- + 2 2 2 2 2 z z z a z z z z 7 2 a z 3 5 7 2 5 z 2 2 > -- + - - 18 a z - 33 a z - 17 a z - 3 a z - 9 z + ---- - 34 a z - z a 2 a 3 4 2 6 2 2 z 3 3 3 5 3 7 3 > 24 a z - 4 a z + ---- + 28 a z + 43 a z + 20 a z + 3 a z + a 4 5 4 4 z 2 4 4 4 6 4 5 z 5 3 5 > 8 z - ---- + 27 a z + 19 a z + 4 a z - ---- - 25 a z - 29 a z - 2 a a 6 7 5 5 6 z 2 6 4 6 6 6 2 z 7 > 9 a z - 8 z + -- - 19 a z - 9 a z + a z + ---- + 6 a z + 2 a a 3 7 5 7 8 2 8 4 8 9 3 9 > 8 a z + 4 a z + 3 z + 7 a z + 4 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	6 7 2 3 3 6 2 5 6 5 -- + - + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 3 q 13 5 11 4 9 4 9 3 7 3 7 2 5 2 5 q q t q t q t q t q t q t q t q t 5 4 t 2 3 2 5 3 5 4 7 4 > ---- + --- + 4 q t + 2 q t + 4 q t + 2 q t + q t + q t 3 q q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n341

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