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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-19/2 + 2q-17/2 - 5q-15/2 + 9q-13/2 - 12q-11/2 + 14q-9/2 - 15q-7/2 + 12q-5/2 - 10q-3/2 + 6q-1/2 - 3q1/2 + q3/2

A2 (sl(3)) Invariant:

q-28 + 2q-24 + q-22 - 2q-20 + 2q-18 - 2q-16 + 2q-14 + 2q-12 + 4q-8 - 2q-6 + 2q-4 - 1 + q2 - q4

HOMFLY-PT Polynomial:

2az + 3az3 + az5 - 2a3z - 5a3z3 - 4a3z5 - a3z7 - a5z-1 - 8a5z - 10a5z3 - 5a5z5 - a5z7 + a7z-1 + 5a7z + 4a7z3 + a7z5

Kauffman Polynomial:

- 2z2 + 3z4 - z6 + 2az - 7az3 + 9az5 - 3az7 + a2z2 - 5a2z4 + 10a2z6 - 4a2z8 + a3z - 2a3z3 + 3a3z5 + 4a3z7 - 3a3z9 + 7a4z2 - 18a4z4 + 18a4z6 - 5a4z8 - a4z10 + a5z-1 - 8a5z + 13a5z3 - 18a5z5 + 15a5z7 - 6a5z9 - a6 + 14a6z2 - 29a6z4 + 19a6z6 - 5a6z8 - a6z10 + a7z-1 - 4a7z + 3a7z3 - 6a7z5 + 5a7z7 - 3a7z9 + 9a8z2 - 15a8z4 + 10a8z6 - 4a8z8 + a9z - 2a9z3 + 5a9z5 - 3a9z7 - a10z2 + 4a10z4 - 2a10z6 - 2a11z + 3a11z3 - 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                     2   j = 0                   4 1   j = -2                 6 2     j = -4               7 5       j = -6             8 5         j = -8           6 7           j = -10         6 8             j = -12       3 6               j = -14     2 6                 j = -16   1 4                   j = -18   1                     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, 376]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 376]]
Out[4]=	PD[X[12, 1, 13, 2], X[14, 3, 15, 4], X[16, 5, 17, 6], X[22, 7, 11, 8], > X[18, 10, 19, 9], X[20, 18, 21, 17], X[10, 20, 1, 19], X[8, 11, 9, 12], > X[6, 21, 7, 22], X[4, 13, 5, 14], X[2, 15, 3, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 2, -10, 3, -9, 4, -8, 5, -7}, > {8, -1, 10, -2, 11, -3, 6, -5, 7, -6, 9, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 2 5 9 12 14 15 12 10 -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 6 3/2 > ------- - 3 Sqrt[q] + q Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 2 -22 2 2 2 2 2 4 2 2 2 4 -1 + q + --- + q - --- + --- - --- + --- + --- + -- - -- + -- + q - q 24 20 18 16 14 12 8 6 4 q q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 376]][a, z]
Out[8]=	5 7 a a 3 5 7 3 3 3 5 3 -(--) + -- + 2 a z - 2 a z - 8 a z + 5 a z + 3 a z - 5 a z - 10 a z + z z 7 3 5 3 5 5 5 7 5 3 7 5 7 > 4 a z + a z - 4 a z - 5 a z + a z - a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 376]][a, z]
Out[9]=	5 7 6 a a 3 5 7 9 11 2 -a + -- + -- + 2 a z + a z - 8 a z - 4 a z + a z - 2 a z - 2 z + z z 2 2 4 2 6 2 8 2 10 2 3 3 3 > a z + 7 a z + 14 a z + 9 a z - a z - 7 a z - 2 a z + 5 3 7 3 9 3 11 3 4 2 4 4 4 > 13 a z + 3 a z - 2 a z + 3 a z + 3 z - 5 a z - 18 a z - 6 4 8 4 10 4 5 3 5 5 5 7 5 > 29 a z - 15 a z + 4 a z + 9 a z + 3 a z - 18 a z - 6 a z + 9 5 11 5 6 2 6 4 6 6 6 8 6 > 5 a z - a z - z + 10 a z + 18 a z + 19 a z + 10 a z - 10 6 7 3 7 5 7 7 7 9 7 2 8 > 2 a z - 3 a z + 4 a z + 15 a z + 5 a z - 3 a z - 4 a z - 4 8 6 8 8 8 3 9 5 9 7 9 4 10 6 10 > 5 a z - 5 a z - 4 a z - 3 a z - 6 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 6 1 1 1 4 2 6 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 6 6 8 6 7 8 5 7 > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + 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 2 t 2 2 2 4 3 > --- + t + 2 q t + q t 2 q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a376

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