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DrawMorseLink

PD Presentation:

X6172 X2536 X18,12,19,11 X10,3,11,4 X4,9,1,10 X14,7,15,8 X8,13,5,14 X22,16,13,15 X20,18,21,17 X16,22,17,21 X12,20,9,19

Gauss Code:

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

Jones Polynomial:

- q-13/2 + q-11/2 - 5q-9/2 + 5q-7/2 - 11q-5/2 + 11q-3/2 - 14q-1/2 + 11q1/2 - 9q3/2 + 7q5/2 - 4q7/2 + q9/2

A2 (sl(3)) Invariant:

q-22 + 3q-20 + 4q-18 + 8q-16 + 12q-14 + 10q-12 + 13q-10 + 12q-8 + 6q-6 + 7q-4 + 2q-2 + 3 + q2 - 2q4 + 2q6 - 2q8 + 2q12 - q14

HOMFLY-PT Polynomial:

a-3z3 - a-1z3 - a-1z5 - az-3 - 3az-1 - 3az - az3 - az5 + 3a3z-3 + 7a3z-1 + 6a3z + 3a3z3 - 3a5z-3 - 5a5z-1 - 3a5z + a7z-3 + a7z-1

Kauffman Polynomial:

2a-4z4 - a-4z6 - 4a-3z3 + 11a-3z5 - 4a-3z7 - 13a-2z4 + 19a-2z6 - 6a-2z8 - 5a-1z5 + 10a-1z7 - 4a-1z9 - 12z4 + 16z6 - 4z8 - z10 - az-3 + 4az-1 - 6az + 8az3 - 17az5 + 14az7 - 5az9 + 3a2z-2 - 8a2 + 6a2z2 + 3a2z4 - 5a2z6 + a2z8 - a2z10 - 3a3z-3 + 9a3z-1 - 14a3z + 12a3z3 - 3a3z5 - a3z7 - a3z9 + 6a4z-2 - 15a4 + 12a4z2 - 2a4z6 - a4z8 - 3a5z-3 + 9a5z-1 - 14a5z + 12a5z3 - 3a5z5 - a5z7 + 3a6z-2 - 8a6 + 6a6z2 - a6z6 - a7z-3 + 4a7z-1 - 6a7z + 4a7z3 - a7z5

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 10                       1 j = 8                     3   j = 6                   4 1   j = 4                 5 3     j = 2               6 4       j = 0             8 5         j = -2           7 10           j = -4         4 4             j = -6       1 7               j = -8     4 4                 j = -10   1 5                   j = -12                         j = -14 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]=	4
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 533]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 533]]
Out[4]=	PD[X[6, 1, 7, 2], X[2, 5, 3, 6], X[18, 12, 19, 11], X[10, 3, 11, 4], > X[4, 9, 1, 10], X[14, 7, 15, 8], X[8, 13, 5, 14], X[22, 16, 13, 15], > X[20, 18, 21, 17], X[16, 22, 17, 21], X[12, 20, 9, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -11}, > {7, -6, 8, -10, 9, -3, 11, -9, 10, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) -(11/2) 5 5 11 11 14 -q + q - ---- + ---- - ---- + ---- - ------- + 11 Sqrt[q] - 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 3/2 5/2 7/2 9/2 > 9 q + 7 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 3 4 8 12 10 13 12 6 7 2 2 4 3 + q + --- + --- + --- + --- + --- + --- + -- + -- + -- + -- + q - 2 q + 20 18 16 14 12 10 8 6 4 2 q q q q q q q q q q 6 8 12 14 > 2 q - 2 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 533]][a, z]
Out[8]=	3 5 7 3 5 7 a 3 a 3 a a 3 a 7 a 5 a a 3 5 -(--) + ---- - ---- + -- - --- + ---- - ---- + -- - 3 a z + 6 a z - 3 a z + 3 3 3 3 z z z z z z z z 3 3 5 z z 3 3 3 z 5 > -- - -- - a z + 3 a z - -- - a z 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 533]][a, z]
Out[9]=	3 5 7 2 4 6 2 4 6 a 3 a 3 a a 3 a 6 a 3 a 4 a -8 a - 15 a - 8 a - -- - ---- - ---- - -- + ---- + ---- + ---- + --- + 3 3 3 3 2 2 2 z z z z z z z z 3 5 7 9 a 9 a 4 a 3 5 7 2 2 > ---- + ---- + ---- - 6 a z - 14 a z - 14 a z - 6 a z + 6 a z + z z z 3 4 2 6 2 4 z 3 3 3 5 3 7 3 > 12 a z + 6 a z - ---- + 8 a z + 12 a z + 12 a z + 4 a z - 3 a 4 4 5 5 4 2 z 13 z 2 4 11 z 5 z 5 3 5 > 12 z + ---- - ----- + 3 a z + ----- - ---- - 17 a z - 3 a z - 4 2 3 a a a a 6 6 7 5 5 7 5 6 z 19 z 2 6 4 6 6 6 4 z > 3 a z - a z + 16 z - -- + ----- - 5 a z - 2 a z - a z - ---- + 4 2 3 a a a 7 8 9 10 z 7 3 7 5 7 8 6 z 2 8 4 8 4 z > ----- + 14 a z - a z - a z - 4 z - ---- + a z - a z - ---- - a 2 a a 9 3 9 10 2 10 > 5 a z - a z - z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	10 1 1 5 4 4 1 7 4 8 + -- + ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 2 14 6 10 5 10 4 8 4 8 3 6 3 6 2 4 2 q q t q t q t q t q t q t q t q t 4 7 2 2 2 4 2 4 3 6 3 > ---- + ---- + 5 t + 6 q t + 4 q t + 5 q t + 3 q t + 4 q t + 4 2 q t q t 6 4 8 4 10 5 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a533

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