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DrawMorseLink

PD Presentation:

X10,1,11,2 X20,11,21,12 X6,9,7,10 X16,7,17,8 X8,15,1,16 X22,17,9,18 X12,4,13,3 X18,6,19,5 X4,14,5,13 X14,21,15,22 X2,20,3,19

Gauss Code:

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

Jones Polynomial:

q-15/2 - 4q-13/2 + 10q-11/2 - 17q-9/2 + 24q-7/2 - 29q-5/2 + 28q-3/2 - 26q-1/2 + 19q1/2 - 12q3/2 + 5q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-22 + 2q-20 - 3q-18 - q-16 + 3q-14 - 6q-12 + 5q-10 + 2q-6 + 6q-4 - 3q-2 + 7 - 3q2 + 3q6 - 3q8 + q10

HOMFLY-PT Polynomial:

- a-1z3 - a-1z5 - 2az-1 - 3az + 2az5 + az7 + 3a3z-1 + 5a3z + 5a3z3 + 3a3z5 + a3z7 - a5z-1 - 2a5z - 2a5z3 - a5z5

Kauffman Polynomial:

- a-3z5 + 3a-2z4 - 5a-2z6 - 5a-1z3 + 15a-1z5 - 12a-1z7 + 2z2 - 9z4 + 23z6 - 16z8 + 2az-1 - 5az - az3 + 9az5 + 8az7 - 12az9 - 3a2 + 8a2z2 - 29a2z4 + 48a2z6 - 19a2z8 - 4a2z10 + 3a3z-1 - 9a3z + 15a3z3 - 26a3z5 + 40a3z7 - 21a3z9 - 3a4 + 12a4z2 - 32a4z4 + 38a4z6 - 11a4z8 - 4a4z10 + a5z-1 - 4a5z + 7a5z3 - 11a5z5 + 16a5z7 - 9a5z9 - a6 + 5a6z2 - 13a6z4 + 17a6z6 - 8a6z8 - 4a7z3 + 8a7z5 - 4a7z7 - a8z2 + 2a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                       1 j = 6                     4   j = 4                   8 1   j = 2                 11 4     j = 0               15 8       j = -2             15 13         j = -4           14 13           j = -6         10 15             j = -8       7 14               j = -10     3 10                 j = -12   1 7                   j = -14   3                     j = -16 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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 317]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 317]]
Out[4]=	PD[X[10, 1, 11, 2], X[20, 11, 21, 12], X[6, 9, 7, 10], X[16, 7, 17, 8], > X[8, 15, 1, 16], X[22, 17, 9, 18], X[12, 4, 13, 3], X[18, 6, 19, 5], > X[4, 14, 5, 13], X[14, 21, 15, 22], X[2, 20, 3, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 7, -9, 8, -3, 4, -5}, > {3, -1, 2, -7, 9, -10, 5, -4, 6, -8, 11, -2, 10, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 4 10 17 24 29 28 26 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 19 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 7/2 > 12 q + 5 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 2 3 -16 3 6 5 2 6 3 2 6 7 - q + --- - --- - q + --- - --- + --- + -- + -- - -- - 3 q + 3 q - 20 18 14 12 10 6 4 2 q q q q q q q q 8 10 > 3 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 317]][a, z]
Out[8]=	3 5 3 5 -2 a 3 a a 3 5 z 3 3 5 3 z ---- + ---- - -- - 3 a z + 5 a z - 2 a z - -- + 5 a z - 2 a z - -- + z z z a a 5 3 5 5 5 7 3 7 > 2 a z + 3 a z - a z + a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 317]][a, z]
Out[9]=	3 5 2 4 6 2 a 3 a a 3 5 2 -3 a - 3 a - a + --- + ---- + -- - 5 a z - 9 a z - 4 a z + 2 z + z z z 3 2 2 4 2 6 2 8 2 5 z 3 3 3 5 3 > 8 a z + 12 a z + 5 a z - a z - ---- - a z + 15 a z + 7 a z - a 4 5 7 3 4 3 z 2 4 4 4 6 4 8 4 z > 4 a z - 9 z + ---- - 29 a z - 32 a z - 13 a z + 2 a z - -- + 2 3 a a 5 6 15 z 5 3 5 5 5 7 5 6 5 z 2 6 > ----- + 9 a z - 26 a z - 11 a z + 8 a z + 23 z - ---- + 48 a z + a 2 a 7 4 6 6 6 8 6 12 z 7 3 7 5 7 > 38 a z + 17 a z - a z - ----- + 8 a z + 40 a z + 16 a z - a 7 7 8 2 8 4 8 6 8 9 3 9 > 4 a z - 16 z - 19 a z - 11 a z - 8 a z - 12 a z - 21 a z - 5 9 2 10 4 10 > 9 a z - 4 a z - 4 a z
In[10]:=	Kh[L][q, t]
Out[10]=	13 1 3 1 7 3 10 7 14 15 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 10 15 14 13 15 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 8 t + 11 q t + 4 q t + 8 q t + 6 3 6 2 4 2 4 2 q t q t q t q t q t 4 3 6 3 8 4 > q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a317

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