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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X18,9,19,10 X8,17,9,18 X4,19,1,20 X14,6,15,5 X10,4,11,3 X20,12,21,11 X22,14,5,13 X12,22,13,21 X2,16,3,15

Gauss Code:

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

Jones Polynomial:

q-9/2 - 3q-7/2 + 5q-5/2 - 8q-3/2 + 10q-1/2 - 13q1/2 + 12q3/2 - 11q5/2 + 8q7/2 - 5q9/2 + 3q11/2 - q13/2

A2 (sl(3)) Invariant:

- q-12 + q-10 - q-8 + 2q-6 + q-4 + q-2 + 4 - q2 + 4q4 - q6 + q8 - 2q12 + q14 - q16 + q18

HOMFLY-PT Polynomial:

- 2a-3z - 7a-3z3 - 5a-3z5 - a-3z7 - a-1z-1 + 4a-1z + 16a-1z3 + 17a-1z5 + 7a-1z7 + a-1z9 + az-1 - 2az - 7az3 - 5az5 - az7

Kauffman Polynomial:

2a-7z3 - a-7z5 - 2a-6z2 + 7a-6z4 - 3a-6z6 - 4a-5z3 + 9a-5z5 - 4a-5z7 - 6a-4z4 + 9a-4z6 - 4a-4z8 - 4a-3z + 16a-3z3 - 24a-3z5 + 14a-3z7 - 4a-3z9 + 5a-2z2 - 11a-2z4 + 4a-2z8 - 2a-2z10 - a-1z-1 - 8a-1z + 44a-1z3 - 65a-1z5 + 36a-1z7 - 8a-1z9 + 1 + 5z2 - 10z4 + 2z6 + 4z8 - 2z10 - az-1 - 4az + 16az3 - 21az5 + 15az7 - 4az9 + a2z2 - 9a2z4 + 13a2z6 - 4a2z8 - 6a3z3 + 10a3z5 - 3a3z7 - a4z2 + 3a4z4 - a4z6

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 14                       1 j = 12                     2   j = 10                   3 1   j = 8                 5 2     j = 6               6 3       j = 4             6 5         j = 2           7 6           j = 0         5 8             j = -2       3 5               j = -4     2 5                 j = -6   1 3                   j = -8   2                     j = -10 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, 51]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 51]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[18, 9, 19, 10], X[8, 17, 9, 18], > X[4, 19, 1, 20], X[14, 6, 15, 5], X[10, 4, 11, 3], X[20, 12, 21, 11], > X[22, 14, 5, 13], X[12, 22, 13, 21], X[2, 16, 3, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 7, -5}, {6, -1, 2, -4, 3, -7, 8, -10, 9, -6, 11, -2, 4, -3, > 5, -8, 10, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 3 5 8 10 3/2 5/2 q - ---- + ---- - ---- + ------- - 13 Sqrt[q] + 12 q - 11 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 13/2 > 8 q - 5 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -10 -8 2 -4 -2 2 4 6 8 12 14 4 - q + q - q + -- + q + q - q + 4 q - q + q - 2 q + q - 6 q 16 18 > q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 51]][a, z]
Out[8]=	3 3 5 5 1 a 2 z 4 z 7 z 16 z 3 5 z 17 z -(---) + - - --- + --- - 2 a z - ---- + ----- - 7 a z - ---- + ----- - a z z 3 a 3 a 3 a a a a 7 7 9 5 z 7 z 7 z > 5 a z - -- + ---- - a z + -- 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 51]][a, z]
Out[9]=	2 2 3 1 a 4 z 8 z 2 2 z 5 z 2 2 4 2 2 z 1 - --- - - - --- - --- - 4 a z + 5 z - ---- + ---- + a z - a z + ---- - a z z 3 a 6 2 7 a a a a 3 3 3 4 4 4 4 z 16 z 44 z 3 3 3 4 7 z 6 z 11 z > ---- + ----- + ----- + 16 a z - 6 a z - 10 z + ---- - ---- - ----- - 5 3 a 6 4 2 a a a a a 5 5 5 5 2 4 4 4 z 9 z 24 z 65 z 5 3 5 6 > 9 a z + 3 a z - -- + ---- - ----- - ----- - 21 a z + 10 a z + 2 z - 7 5 3 a a a a 6 6 7 7 7 3 z 9 z 2 6 4 6 4 z 14 z 36 z 7 3 7 > ---- + ---- + 13 a z - a z - ---- + ----- + ----- + 15 a z - 3 a z + 6 4 5 3 a a a a a 8 8 9 9 10 8 4 z 4 z 2 8 4 z 8 z 9 10 2 z > 4 z - ---- + ---- - 4 a z - ---- - ---- - 4 a z - 2 z - ----- 4 2 3 a 2 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 3 2 5 3 5 5 8 + 7 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + ---- + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t q t q t 2 4 4 2 6 2 6 3 8 3 8 4 > 6 q t + 6 q t + 5 q t + 6 q t + 3 q t + 5 q t + 2 q t + 10 4 10 5 12 5 14 6 > 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a51

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