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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-5 + 4q-4 - 9q-3 + 16q-2 - 21q-1 + 25 - 24q + 22q2 - 15q3 + 10q4 - 4q5 + q6

A2 (sl(3)) Invariant:

- q-14 + 2q-12 - 3q-10 + 3q-8 + 2q-6 - 3q-4 + 6q-2 - 3 + 6q2 + 3q4 + 3q6 + 8q8 - q10 + 4q12 + q14 - q16 + q18

HOMFLY-PT Polynomial:

a-4z-2 + 2a-4 + 2a-4z2 + a-4z4 - 2a-2z-2 - 5a-2 - 9a-2z2 - 7a-2z4 - 2a-2z6 + z-2 + 3 + 8z2 + 10z4 + 5z6 + z8 - 3a2z2 - 3a2z4 - a2z6

Kauffman Polynomial:

a-6z2 - 2a-6z4 + a-6z6 + 3a-5z3 - 8a-5z5 + 4a-5z7 - a-4z-2 + 6a-4 - 12a-4z2 + 16a-4z4 - 19a-4z6 + 8a-4z8 + 2a-3z-1 - 6a-3z + 4a-3z3 + 3a-3z5 - 14a-3z7 + 8a-3z9 - 2a-2z-2 + 12a-2 - 37a-2z2 + 62a-2z4 - 52a-2z6 + 13a-2z8 + 3a-2z10 + 2a-1z-1 - 8a-1z + 5a-1z3 + 21a-1z5 - 36a-1z7 + 17a-1z9 - z-2 + 8 - 32z2 + 64z4 - 55z6 + 16z8 + 3z10 - 3az + 9az3 - 2az5 - 10az7 + 9az9 + a2 - 7a2z2 + 15a2z4 - 19a2z6 + 11a2z8 - a3z + 4a3z3 - 11a3z5 + 8a3z7 + a4z2 - 5a4z4 + 4a4z6 - a5z3 + a5z5

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 = 13                       1 j = 11                     3   j = 9                   7 1   j = 7                 9 4     j = 5               13 6       j = 3             12 10         j = 1           13 12           j = -1         9 13             j = -3       7 12               j = -5     3 10                 j = -7   1 6                   j = -9   3                     j = -11 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 522]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 522]]
Out[4]=	PD[X[8, 1, 9, 2], X[14, 4, 15, 3], X[16, 5, 17, 6], X[18, 8, 19, 7], > X[12, 20, 7, 19], X[10, 14, 11, 13], X[22, 10, 13, 9], X[20, 15, 21, 16], > X[6, 17, 1, 18], X[2, 11, 3, 12], X[4, 22, 5, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11, 3, -9}, {4, -1, 7, -6, 10, -5}, > {6, -2, 8, -3, 9, -4, 5, -8, 11, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-5 4 9 16 21 2 3 4 5 6 25 - q + -- - -- + -- - -- - 24 q + 22 q - 15 q + 10 q - 4 q + q 4 3 2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 2 3 3 2 3 6 2 4 6 8 10 -3 - q + --- - --- + -- + -- - -- + -- + 6 q + 3 q + 3 q + 8 q - q + 12 10 8 6 4 2 q q q q q q 12 14 16 18 > 4 q + q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 522]][a, z]
Out[8]=	2 2 4 2 5 -2 1 2 2 2 z 9 z 2 2 4 z 3 + -- - -- + z + ----- - ----- + 8 z + ---- - ---- - 3 a z + 10 z + -- - 4 2 4 2 2 2 4 2 4 a a a z a z a a a 4 6 7 z 2 4 6 2 z 2 6 8 > ---- - 3 a z + 5 z - ---- - a z + z 2 2 a a
In[9]:=	Kauffman[Link[11, Alternating, 522]][a, z]
Out[9]=	6 12 2 -2 1 2 2 2 6 z 8 z 8 + -- + -- + a - z - ----- - ----- + ---- + --- - --- - --- - 3 a z - 4 2 4 2 2 2 3 a z 3 a a a a z a z a z a 2 2 2 3 3 3 3 2 z 12 z 37 z 2 2 4 2 3 z 4 z 5 z > a z - 32 z + -- - ----- - ----- - 7 a z + a z + ---- + ---- + ---- + 6 4 2 5 3 a a a a a a 4 4 4 3 3 3 5 3 4 2 z 16 z 62 z 2 4 > 9 a z + 4 a z - a z + 64 z - ---- + ----- + ----- + 15 a z - 6 4 2 a a a 5 5 5 6 4 4 8 z 3 z 21 z 5 3 5 5 5 6 z > 5 a z - ---- + ---- + ----- - 2 a z - 11 a z + a z - 55 z + -- - 5 3 a 6 a a a 6 6 7 7 7 19 z 52 z 2 6 4 6 4 z 14 z 36 z 7 > ----- - ----- - 19 a z + 4 a z + ---- - ----- - ----- - 10 a z + 4 2 5 3 a a a a a 8 8 9 9 3 7 8 8 z 13 z 2 8 8 z 17 z 9 10 > 8 a z + 16 z + ---- + ----- + 11 a z + ---- + ----- + 9 a z + 3 z + 4 2 3 a a a a 10 3 z > ----- 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	13 1 3 1 6 3 10 7 12 -- + 13 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t 9 3 3 2 5 2 5 3 7 3 > --- + 12 q t + 12 q t + 10 q t + 13 q t + 6 q t + 9 q t + q t 7 4 9 4 9 5 11 5 13 6 > 4 q t + 7 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a522

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