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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-17/2 - 3q-15/2 + 6q-13/2 - 9q-11/2 + 11q-9/2 - 13q-7/2 + 12q-5/2 - 10q-3/2 + 7q-1/2 - 5q1/2 + 2q3/2 - q5/2

A2 (sl(3)) Invariant:

- q-26 + q-22 - q-20 + 2q-18 - q-16 + q-12 - 2q-10 + 3q-8 - q-6 + 2q-4 + 2q-2 + 2q2 + q6 + q8

HOMFLY-PT Polynomial:

- a-1z-1 - 3a-1z - a-1z3 + az-1 + 5az + 7az3 + 2az5 - 3a3z - 5a3z3 - 4a3z5 - a3z7 + 3a5z + 6a5z3 + 2a5z5 - 2a7z - a7z3

Kauffman Polynomial:

- a-1z-1 + 5a-1z - 8a-1z3 + 5a-1z5 - a-1z7 + 1 + z2 - 8z4 + 8z6 - 2z8 - az-1 + 6az - 11az3 + 3az5 + 5az7 - 2az9 - 9a2z4 + 11a2z6 - a2z8 - a2z10 + 3a3z3 - 10a3z5 + 14a3z7 - 5a3z9 + 5a4z2 - 15a4z4 + 16a4z6 - 4a4z8 - a4z10 + 2a5z - 6a5z3 + 5a5z5 + 2a5z7 - 3a5z9 + 3a6z2 - 7a6z4 + 8a6z6 - 5a6z8 + 3a7z - 9a7z3 + 10a7z5 - 6a7z7 - 2a8z2 + 6a8z4 - 5a8z6 + 3a9z3 - 3a9z5 + a10z2 - a10z4

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 = 6                       1 j = 4                     1   j = 2                   4 1   j = 0                 3 1     j = -2               7 4       j = -4             7 5         j = -6           6 5           j = -8         5 7             j = -10       4 6               j = -12     2 5                 j = -14   1 4                   j = -16   2                     j = -18 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, 312]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 312]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[8, 9, 1, 10], > X[22, 13, 9, 14], X[14, 8, 15, 7], X[18, 6, 19, 5], X[20, 17, 21, 18], > X[16, 21, 17, 22], X[6, 16, 7, 15], X[4, 20, 5, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -11, 7, -10, 6, -4}, > {4, -1, 2, -3, 5, -6, 10, -9, 8, -7, 11, -8, 9, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 3 6 9 11 13 12 10 7 q - ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q 3/2 5/2 > 5 Sqrt[q] + 2 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 -22 -20 2 -16 -12 2 3 -6 2 2 2 -q + q - q + --- - q + q - --- + -- - q + -- + -- + 2 q + 18 10 8 4 2 q q q q q 6 8 > q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 312]][a, z]
Out[8]=	3 1 a 3 z 3 5 7 z 3 3 3 -(---) + - - --- + 5 a z - 3 a z + 3 a z - 2 a z - -- + 7 a z - 5 a z + a z z a a 5 3 7 3 5 3 5 5 5 3 7 > 6 a z - a z + 2 a z - 4 a z + 2 a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 312]][a, z]
Out[9]=	1 a 5 z 5 7 2 4 2 6 2 1 - --- - - + --- + 6 a z + 2 a z + 3 a z + z + 5 a z + 3 a z - a z z a 3 8 2 10 2 8 z 3 3 3 5 3 7 3 9 3 > 2 a z + a z - ---- - 11 a z + 3 a z - 6 a z - 9 a z + 3 a z - a 5 4 2 4 4 4 6 4 8 4 10 4 5 z 5 > 8 z - 9 a z - 15 a z - 7 a z + 6 a z - a z + ---- + 3 a z - a 3 5 5 5 7 5 9 5 6 2 6 4 6 > 10 a z + 5 a z + 10 a z - 3 a z + 8 z + 11 a z + 16 a z + 7 6 6 8 6 z 7 3 7 5 7 7 7 8 > 8 a z - 5 a z - -- + 5 a z + 14 a z + 2 a z - 6 a z - 2 z - a 2 8 4 8 6 8 9 3 9 5 9 2 10 4 10 > a z - 4 a z - 5 a z - 2 a z - 5 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 7 1 2 1 4 2 5 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 18 7 16 6 14 6 14 5 12 5 12 4 10 4 q q q t q t q t q t q t q t q t 6 5 7 6 5 7 4 t 2 2 2 > ------ + ----- + ----- + ----- + ---- + ---- + 3 t + --- + t + 4 q t + 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q 2 3 4 3 6 4 > q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a312

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