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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 4q-11/2 - 9q-9/2 + 13q-7/2 - 19q-5/2 + 21q-3/2 - 21q-1/2 + 18q1/2 - 13q3/2 + 8q5/2 - 4q7/2 + q9/2

A2 (sl(3)) Invariant:

q-20 - 2q-18 + q-16 + 5q-14 - 2q-12 + 4q-10 + 2q-8 - 2q-6 + 3q-4 - 3q-2 + 3 - q2 - 2q4 + 4q6 - 3q8 + 2q12 - q14

HOMFLY-PT Polynomial:

a-3z3 - a-1z5 - az - 2az3 - 2az5 - a3z-1 - a3z - a3z5 + a5z-1 + a5z3

Kauffman Polynomial:

2a-4z4 - a-4z6 - 5a-3z3 + 10a-3z5 - 4a-3z7 + 4a-2z2 - 15a-2z4 + 19a-2z6 - 7a-2z8 - a-1z3 - 3a-1z5 + 11a-1z7 - 6a-1z9 + 12z2 - 36z4 + 36z6 - 10z8 - 2z10 - az + 4az3 - 18az5 + 25az7 - 12az9 + 11a2z2 - 34a2z4 + 35a2z6 - 12a2z8 - 2a2z10 - a3z-1 + 2a3z - 9a3z3 + 10a3z5 + 2a3z7 - 6a3z9 + a4 + 3a4z2 - 10a4z4 + 15a4z6 - 9a4z8 - a5z-1 + 3a5z - 8a5z3 + 14a5z5 - 8a5z7 + 5a6z4 - 4a6z6 + a7z3 - 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                   5 1   j = 4                 8 3     j = 2               10 5       j = 0             11 8         j = -2           11 11           j = -4         8 10             j = -6       5 11               j = -8     4 8                 j = -10   1 6                   j = -12   3                     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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 138]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 138]]
Out[4]=	PD[X[6, 1, 7, 2], X[2, 11, 3, 12], X[14, 3, 15, 4], X[12, 5, 13, 6], > X[22, 13, 5, 14], X[4, 21, 1, 22], X[20, 16, 21, 15], X[16, 8, 17, 7], > X[8, 20, 9, 19], X[18, 10, 19, 9], X[10, 18, 11, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -6}, {4, -1, 8, -9, 10, -11, 2, -4, 5, -3, 7, -8, 11, -10, > 9, -7, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 4 9 13 19 21 21 -q + ----- - ---- + ---- - ---- + ---- - ------- + 18 Sqrt[q] - 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 3/2 5/2 7/2 9/2 > 13 q + 8 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 2 -16 5 2 4 2 2 3 3 2 4 3 + q - --- + q + --- - --- + --- + -- - -- + -- - -- - q - 2 q + 18 14 12 10 8 6 4 2 q q q q q q q q 6 8 12 14 > 4 q - 3 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 138]][a, z]
Out[8]=	3 5 3 5 a a 3 z 3 5 3 z 5 3 5 -(--) + -- - a z - a z + -- - 2 a z + a z - -- - 2 a z - a z z z 3 a a
In[9]:=	Kauffman[Link[11, Alternating, 138]][a, z]
Out[9]=	3 5 2 4 a a 3 5 2 4 z 2 2 4 2 a - -- - -- - a z + 2 a z + 3 a z + 12 z + ---- + 11 a z + 3 a z - z z 2 a 3 3 4 4 5 z z 3 3 3 5 3 7 3 4 2 z 15 z > ---- - -- + 4 a z - 9 a z - 8 a z + a z - 36 z + ---- - ----- - 3 a 4 2 a a a 5 5 2 4 4 4 6 4 10 z 3 z 5 3 5 > 34 a z - 10 a z + 5 a z + ----- - ---- - 18 a z + 10 a z + 3 a a 6 6 5 5 7 5 6 z 19 z 2 6 4 6 6 6 > 14 a z - a z + 36 z - -- + ----- + 35 a z + 15 a z - 4 a z - 4 2 a a 7 7 8 4 z 11 z 7 3 7 5 7 8 7 z 2 8 > ---- + ----- + 25 a z + 2 a z - 8 a z - 10 z - ---- - 12 a z - 3 a 2 a a 9 4 8 6 z 9 3 9 10 2 10 > 9 a z - ---- - 12 a z - 6 a z - 2 z - 2 a z a
In[10]:=	Kh[L][q, t]
Out[10]=	11 1 3 1 6 4 8 5 11 11 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 8 10 11 2 2 2 4 2 4 3 > ----- + ---- + ---- + 8 t + 10 q t + 5 q t + 8 q t + 3 q t + 4 2 4 2 q t q t q t 6 3 6 4 8 4 10 5 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a138

L11a137 L11a139