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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X18,12,19,11 X12,6,13,5 X4,19,5,20 X14,7,15,8 X20,13,21,14 X22,15,7,16 X16,21,17,22 X6,18,1,17

Gauss Code:

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

Jones Polynomial:

- q-19/2 + 3q-17/2 - 6q-15/2 + 11q-13/2 - 15q-11/2 + 17q-9/2 - 18q-7/2 + 15q-5/2 - 12q-3/2 + 7q-1/2 - 4q1/2 + q3/2

A2 (sl(3)) Invariant:

q-28 - q-26 + q-24 - 3q-20 + 2q-18 - 3q-16 + 2q-14 + 2q-12 + 5q-8 - 2q-6 + 3q-4 + q-2 + 2q2 - q4

HOMFLY-PT Polynomial:

- az-1 - az + 2az3 + az5 + a3z-1 + 2a3z - a3z3 - 3a3z5 - a3z7 - 4a5z - 6a5z3 - 4a5z5 - a5z7 + 2a7z + 3a7z3 + a7z5

Kauffman Polynomial:

2z4 - z6 + az-1 - az - 6az3 + 11az5 - 4az7 - a2 + 2a2z2 - 10a2z4 + 16a2z6 - 6a2z8 + a3z-1 - 4a3z + 4a3z3 - 4a3z5 + 10a3z7 - 5a3z9 + 8a4z2 - 21a4z4 + 20a4z6 - 4a4z8 - 2a4z10 - 5a5z + 23a5z3 - 38a5z5 + 29a5z7 - 10a5z9 + 12a6z2 - 28a6z4 + 19a6z6 - 4a6z8 - 2a6z10 - a7z + 5a7z3 - 12a7z5 + 10a7z7 - 5a7z9 + 4a8z2 - 13a8z4 + 13a8z6 - 6a8z8 + a9z - 6a9z3 + 10a9z5 - 5a9z7 - 2a10z2 + 6a10z4 - 3a10z6 + 2a11z3 - a11z5

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 4                       1 j = 2                     3   j = 0                   4 1   j = -2                 8 3     j = -4               8 5       j = -6             10 7         j = -8           8 9           j = -10         7 9             j = -12       4 8               j = -14     2 7                 j = -16   1 4                   j = -18   2                     j = -20 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, 145]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 145]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[18, 12, 19, 11], > X[12, 6, 13, 5], X[4, 19, 5, 20], X[14, 7, 15, 8], X[20, 13, 21, 14], > X[22, 15, 7, 16], X[16, 21, 17, 22], X[6, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -6, 5, -11}, > {7, -1, 2, -3, 4, -5, 8, -7, 9, -10, 11, -4, 6, -8, 10, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 3 6 11 15 17 18 15 12 -q + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q 7 3/2 > ------- - 4 Sqrt[q] + q Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 -26 -24 3 2 3 2 2 5 2 3 -2 2 q - q + q - --- + --- - --- + --- + --- + -- - -- + -- + q + 2 q - 20 18 16 14 12 8 6 4 q q q q q q q q 4 > q
In[8]:=	HOMFLYPT[Link[11, Alternating, 145]][a, z]
Out[8]=	3 a a 3 5 7 3 3 3 5 3 -(-) + -- - a z + 2 a z - 4 a z + 2 a z + 2 a z - a z - 6 a z + z z 7 3 5 3 5 5 5 7 5 3 7 5 7 > 3 a z + a z - 3 a z - 4 a z + a z - a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 145]][a, z]
Out[9]=	3 2 a a 3 5 7 9 2 2 4 2 -a + - + -- - a z - 4 a z - 5 a z - a z + a z + 2 a z + 8 a z + z z 6 2 8 2 10 2 3 3 3 5 3 7 3 > 12 a z + 4 a z - 2 a z - 6 a z + 4 a z + 23 a z + 5 a z - 9 3 11 3 4 2 4 4 4 6 4 8 4 > 6 a z + 2 a z + 2 z - 10 a z - 21 a z - 28 a z - 13 a z + 10 4 5 3 5 5 5 7 5 9 5 11 5 > 6 a z + 11 a z - 4 a z - 38 a z - 12 a z + 10 a z - a z - 6 2 6 4 6 6 6 8 6 10 6 7 > z + 16 a z + 20 a z + 19 a z + 13 a z - 3 a z - 4 a z + 3 7 5 7 7 7 9 7 2 8 4 8 6 8 > 10 a z + 29 a z + 10 a z - 5 a z - 6 a z - 4 a z - 4 a z - 8 8 3 9 5 9 7 9 4 10 6 10 > 6 a z - 5 a z - 10 a z - 5 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 8 1 2 1 4 2 7 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 20 8 18 7 16 7 16 6 14 6 14 5 12 5 q q q t q t q t q t q t q t q t 8 7 9 8 9 10 7 8 > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + 4 t + 12 4 10 4 10 3 8 3 8 2 6 2 6 4 q t q t q t q t q t q t q t q t 3 t 2 2 2 4 3 > --- + t + 3 q t + q t 2 q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a145

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