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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-5 + 5q-4 - 12q-3 + 21q-2 - 27q-1 + 33 - 31q + 28q2 - 20q3 + 12q4 - 5q5 + q6

A2 (sl(3)) Invariant:

- q-14 + 3q-12 - 4q-10 + 4q-8 + 3q-6 - 2q-4 + 11q-2 - 1 + 9q2 + 2q4 + 6q8 - 5q10 + 3q12 - 2q16 + q18

HOMFLY-PT Polynomial:

a-4z2 + a-4z4 + a-2z-2 + a-2 - 3a-2z2 - 5a-2z4 - 2a-2z6 - 2z-2 - 2 + 2z2 + 6z4 + 4z6 + z8 + a2z-2 + a2 - a2z2 - 2a2z4 - a2z6

Kauffman Polynomial:

- a-6z4 + a-6z6 + 3a-5z3 - 8a-5z5 + 5a-5z7 - 4a-4z2 + 14a-4z4 - 22a-4z6 + 11a-4z8 + 4a-3z3 + a-3z5 - 18a-3z7 + 12a-3z9 + a-2z-2 - 2a-2 - 11a-2z2 + 49a-2z4 - 62a-2z6 + 18a-2z8 + 5a-2z10 - 2a-1z-1 + 2a-1z + 2a-1z3 + 19a-1z5 - 50a-1z7 + 27a-1z9 + 2z-2 - 3 - 10z2 + 53z4 - 72z6 + 25z8 + 5z10 - 2az-1 + 2az + 4az3 - 4az5 - 15az7 + 15az9 + a2z-2 - 2a2 - 3a2z2 + 16a2z4 - 28a2z6 + 18a2z8 + 3a3z3 - 13a3z5 + 12a3z7 - 3a4z4 + 5a4z6 + 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                     4   j = 9                   8 1   j = 7                 12 4     j = 5               16 8       j = 3             16 13         j = 1           17 15           j = -1         12 18             j = -3       9 15               j = -5     4 13                 j = -7   1 8                   j = -9   4                     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, 509]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 509]]
Out[4]=	PD[X[8, 1, 9, 2], X[20, 10, 21, 9], X[14, 5, 15, 6], X[12, 14, 7, 13], > X[16, 8, 17, 7], X[22, 18, 13, 17], X[10, 4, 11, 3], X[18, 11, 19, 12], > X[6, 15, 1, 16], X[4, 20, 5, 19], X[2, 21, 3, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 7, -10, 3, -9}, {5, -1, 2, -7, 8, -4}, > {4, -3, 9, -5, 6, -8, 10, -2, 11, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-5 5 12 21 27 2 3 4 5 6 33 - q + -- - -- + -- - -- - 31 q + 28 q - 20 q + 12 q - 5 q + q 4 3 2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 3 4 4 3 2 11 2 4 8 10 -1 - q + --- - --- + -- + -- - -- + -- + 9 q + 2 q + 6 q - 5 q + 12 10 8 6 4 2 q q q q q q 12 16 18 > 3 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 509]][a, z]
Out[8]=	2 2 2 4 4 -2 2 2 1 a 2 z 3 z 2 2 4 z 5 z -2 + a + a - -- + ----- + -- + 2 z + -- - ---- - a z + 6 z + -- - ---- - 2 2 2 2 4 2 4 2 z a z z a a a a 6 2 4 6 2 z 2 6 8 > 2 a z + 4 z - ---- - a z + z 2 a
In[9]:=	Kauffman[Link[11, Alternating, 509]][a, z]
Out[9]=	2 2 2 2 2 1 a 2 2 a 2 z 2 4 z -3 - -- - 2 a + -- + ----- + -- - --- - --- + --- + 2 a z - 10 z - ---- - 2 2 2 2 2 a z z a 4 a z a z z a 2 3 3 3 4 11 z 2 2 3 z 4 z 2 z 3 3 3 4 z > ----- - 3 a z + ---- + ---- + ---- + 4 a z + 3 a z + 53 z - -- + 2 5 3 a 6 a a a a 4 4 5 5 5 14 z 49 z 2 4 4 4 8 z z 19 z 5 > ----- + ----- + 16 a z - 3 a z - ---- + -- + ----- - 4 a z - 4 2 5 3 a a a a a 6 6 6 7 3 5 5 5 6 z 22 z 62 z 2 6 4 6 5 z > 13 a z + a z - 72 z + -- - ----- - ----- - 28 a z + 5 a z + ---- - 6 4 2 5 a a a a 7 7 8 8 18 z 50 z 7 3 7 8 11 z 18 z 2 8 > ----- - ----- - 15 a z + 12 a z + 25 z + ----- + ----- + 18 a z + 3 a 4 2 a a a 9 9 10 12 z 27 z 9 10 5 z > ----- + ----- + 15 a z + 5 z + ----- 3 a 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	18 1 4 1 8 4 13 9 15 -- + 17 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 12 3 3 2 5 2 5 3 7 3 > --- + 15 q t + 16 q t + 13 q t + 16 q t + 8 q t + 12 q t + q t 7 4 9 4 9 5 11 5 13 6 > 4 q t + 8 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a509

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