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DrawMorseLink

PD Presentation:

X6172 X12,7,13,8 X4,13,1,14 X18,6,19,5 X8493 X20,9,21,10 X10,19,11,20 X14,18,15,17 X22,16,17,15 X16,22,5,21 X2,12,3,11

Gauss Code:

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

Jones Polynomial:

- q-5 + 4q-4 - 9q-3 + 15q-2 - 20q-1 + 24 - 22q + 21q2 - 14q3 + 9q4 - 4q5 + q6

A2 (sl(3)) Invariant:

- q-16 + q-14 + 2q-12 - 4q-10 + 2q-8 - 3q-4 + 6q-2 + 7q2 + 4q4 + 3q6 + 8q8 - 2q10 + 3q12 + 2q14 - 2q16 + q18

HOMFLY-PT Polynomial:

a-4z-2 + a-4 + a-4z2 + a-4z4 - 2a-2z-2 - 4a-2 - 4a-2z2 - 2a-2z4 - a-2z6 + z-2 + 4 + 3z2 - z6 - a2 + a2z2 + 2a2z4 - a4z2

Kauffman Polynomial:

- 2a-6z4 + a-6z6 + 4a-5z3 - 9a-5z5 + 4a-5z7 - a-4z-2 + 3a-4 - 9a-4z2 + 21a-4z4 - 22a-4z6 + 8a-4z8 + 2a-3z-1 - 2a-3z + 4a-3z3 + 2a-3z5 - 12a-3z7 + 7a-3z9 - 2a-2z-2 + 7a-2 - 27a-2z2 + 54a-2z4 - 50a-2z6 + 15a-2z8 + 2a-2z10 + 2a-1z-1 - 4a-1z + 6a-1z3 + 3a-1z5 - 20a-1z7 + 13a-1z9 - z-2 + 7 - 23z2 + 39z4 - 41z6 + 16z8 + 2z10 - 3az + 14az3 - 21az5 + 4az7 + 6az9 + 2a2 - 3a2z2 + 3a2z4 - 10a2z6 + 9a2z8 - a3z + 7a3z3 - 12a3z5 + 8a3z7 + 2a4z2 - 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                   6 1   j = 7                 8 3     j = 5               13 6       j = 3             11 10         j = 1           13 11           j = -1         9 13             j = -3       6 11               j = -5     3 9                 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, 471]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 471]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 7, 13, 8], X[4, 13, 1, 14], X[18, 6, 19, 5], > X[8, 4, 9, 3], X[20, 9, 21, 10], X[10, 19, 11, 20], X[14, 18, 15, 17], > X[22, 16, 17, 15], X[16, 22, 5, 21], X[2, 12, 3, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {8, -4, 7, -6, 10, -9}, > {4, -1, 2, -5, 6, -7, 11, -2, 3, -8, 9, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-5 4 9 15 20 2 3 4 5 6 24 - q + -- - -- + -- - -- - 22 q + 21 q - 14 q + 9 q - 4 q + q 4 3 2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -14 2 4 2 3 6 2 4 6 8 10 -q + q + --- - --- + -- - -- + -- + 7 q + 4 q + 3 q + 8 q - 2 q + 12 10 8 4 2 q q q q q 12 14 16 18 > 3 q + 2 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 471]][a, z]
Out[8]=	2 2 -4 4 2 -2 1 2 2 z 4 z 2 2 4 2 4 + a - -- - a + z + ----- - ----- + 3 z + -- - ---- + a z - a z + 2 4 2 2 2 4 2 a a z a z a a 4 4 6 z 2 z 2 4 6 z > -- - ---- + 2 a z - z - -- 4 2 2 a a a
In[9]:=	Kauffman[Link[11, Alternating, 471]][a, z]
Out[9]=	3 7 2 -2 1 2 2 2 2 z 4 z 7 + -- + -- + 2 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 3 3 3 3 2 9 z 27 z 2 2 4 2 4 z 4 z 6 z > a z - 23 z - ---- - ----- - 3 a z + 2 a z + ---- + ---- + ---- + 4 2 5 3 a a a a a 4 4 4 3 3 3 5 3 4 2 z 21 z 54 z 2 4 > 14 a z + 7 a z - a z + 39 z - ---- + ----- + ----- + 3 a z - 6 4 2 a a a 5 5 5 6 4 4 9 z 2 z 3 z 5 3 5 5 5 6 z > 5 a z - ---- + ---- + ---- - 21 a z - 12 a z + a z - 41 z + -- - 5 3 a 6 a a a 6 6 7 7 7 22 z 50 z 2 6 4 6 4 z 12 z 20 z 7 > ----- - ----- - 10 a z + 4 a z + ---- - ----- - ----- + 4 a z + 4 2 5 3 a a a a a 8 8 9 9 3 7 8 8 z 15 z 2 8 7 z 13 z 9 10 > 8 a z + 16 z + ---- + ----- + 9 a z + ---- + ----- + 6 a z + 2 z + 4 2 3 a a a a 10 2 z > ----- 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	13 1 3 1 6 3 9 6 11 -- + 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 > --- + 11 q t + 11 q t + 10 q t + 13 q t + 6 q t + 8 q t + q t 7 4 9 4 9 5 11 5 13 6 > 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a471

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