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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X9,20,10,21 X7,16,8,17 X13,18,14,19 X19,14,20,15 X15,22,16,11 X17,10,18,5 X21,8,22,9 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

- q-13 + 2q-12 - 3q-11 + 3q-10 - 2q-9 + 3q-8 + q-6 + q-5 - q-4 + q-3

A2 (sl(3)) Invariant:

- q-42 - 3q-40 - q-38 - q-36 - q-34 + 3q-32 + 3q-30 + 6q-28 + 6q-26 + 6q-24 + 5q-22 + q-20 + 2q-18 + q-16 + q-10

HOMFLY-PT Polynomial:

a6 + 5a6z2 + 5a6z4 + a6z6 + 2a8z-2 + 7a8 + 9a8z2 + 6a8z4 + a8z6 - 5a10z-2 - 13a10 - 9a10z2 - a10z4 + 4a12z-2 + 5a12 - a14z-2

Kauffman Polynomial:

- a6 + 5a6z2 - 5a6z4 + a6z6 + 4a7z3 - 5a7z5 + a7z7 - 2a8z-2 + 8a8 - 15a8z2 + 14a8z4 - 7a8z6 + a8z8 + 5a9z-1 - 18a9z + 27a9z3 - 15a9z5 + 2a9z7 - 5a10z-2 + 20a10 - 42a10z2 + 44a10z4 - 21a10z6 + 3a10z8 + 9a11z-1 - 33a11z + 42a11z3 - 16a11z5 - 2a11z7 + a11z9 - 4a12z-2 + 15a12 - 29a12z2 + 38a12z4 - 23a12z6 + 4a12z8 + 5a13z-1 - 19a13z + 26a13z3 - 11a13z5 - 2a13z7 + a13z9 - a14z-2 + 3a14 - 7a14z2 + 13a14z4 - 10a14z6 + 2a14z8 + a15z-1 - 4a15z + 7a15z3 - 5a15z5 + a15z7

Khovanov Homology:

trqj r = -11 r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -5                       1 j = -7                     1 1 j = -9                 1 1     j = -11               2 1 1     j = -13             2 4 1       j = -15           3 1 1         j = -17         2 5 2           j = -19       2 2 1             j = -21     1 2 1               j = -23   1 2                   j = -25   1                     j = -27 1

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, 313]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 313]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[9, 20, 10, 21], X[7, 16, 8, 17], > X[13, 18, 14, 19], X[19, 14, 20, 15], X[15, 22, 16, 11], X[17, 10, 18, 5], > X[21, 8, 22, 9], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -4, 9, -3, 8}, > {11, -2, -5, 6, -7, 4, -8, 5, -6, 3, -9, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-13 2 3 3 2 3 -6 -5 -4 -3 -q + --- - --- + --- - -- + -- + q + q - q + q 12 11 10 9 8 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 3 -38 -36 -34 3 3 6 6 6 5 -20 -q - --- - q - q - q + --- + --- + --- + --- + --- + --- + q + 40 32 30 28 26 24 22 q q q q q q q 2 -16 -10 > --- + q + q 18 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 313]][a, z]
Out[8]=	8 10 12 14 6 8 10 12 2 a 5 a 4 a a 6 2 8 2 a + 7 a - 13 a + 5 a + ---- - ----- + ----- - --- + 5 a z + 9 a z - 2 2 2 2 z z z z 10 2 6 4 8 4 10 4 6 6 8 6 > 9 a z + 5 a z + 6 a z - a z + a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 313]][a, z]
Out[9]=	8 10 12 14 9 6 8 10 12 14 2 a 5 a 4 a a 5 a -a + 8 a + 20 a + 15 a + 3 a - ---- - ----- - ----- - --- + ---- + 2 2 2 2 z z z z z 11 13 15 9 a 5 a a 9 11 13 15 6 2 > ----- + ----- + --- - 18 a z - 33 a z - 19 a z - 4 a z + 5 a z - z z z 8 2 10 2 12 2 14 2 7 3 9 3 > 15 a z - 42 a z - 29 a z - 7 a z + 4 a z + 27 a z + 11 3 13 3 15 3 6 4 8 4 10 4 > 42 a z + 26 a z + 7 a z - 5 a z + 14 a z + 44 a z + 12 4 14 4 7 5 9 5 11 5 13 5 > 38 a z + 13 a z - 5 a z - 15 a z - 16 a z - 11 a z - 15 5 6 6 8 6 10 6 12 6 14 6 7 7 > 5 a z + a z - 7 a z - 21 a z - 23 a z - 10 a z + a z + 9 7 11 7 13 7 15 7 8 8 10 8 12 8 > 2 a z - 2 a z - 2 a z + a z + a z + 3 a z + 4 a z + 14 8 11 9 13 9 > 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-7 -5 1 1 1 2 1 2 2 q + q + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 27 11 25 10 23 10 23 9 21 9 21 8 19 8 q t q t q t q t q t q t q t 1 2 2 1 5 3 2 1 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 21 7 19 7 17 7 19 6 17 6 15 6 17 5 15 5 q t q t q t q t q t q t q t q t 2 1 4 2 1 1 1 1 > ------ + ------ + ------ + ------ + ------ + ------ + ----- + ------ + 13 5 15 4 13 4 11 4 13 3 11 3 9 3 11 2 q t q t q t q t q t q t q t q t 1 1 > ----- + ---- 9 2 7 q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n313

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