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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 5q-11/2 - 11q-9/2 + 17q-7/2 - 24q-5/2 + 27q-3/2 - 27q-1/2 + 23q1/2 - 17q3/2 + 10q5/2 - 5q7/2 + q9/2

A2 (sl(3)) Invariant:

q-20 - 2q-18 - q-16 + 4q-14 - 4q-12 + 4q-10 + 3q-8 - 2q-6 + 5q-4 - 5q-2 + 3 - 2q2 - 2q4 + 6q6 - 2q8 + 2q10 + 2q12 - q14

HOMFLY-PT Polynomial:

- a-3z-1 + a-3z3 + 2a-1z-1 - 3a-1z3 - 2a-1z5 - 2az-1 + az + 5az3 + 3az5 + az7 + a3z-1 - a3z - 3a3z3 - 2a3z5 + a5z3

Kauffman Polynomial:

a-4z4 - a-4z6 + a-3z-1 - a-3z - 5a-3z3 + 10a-3z5 - 5a-3z7 + 2a-2z2 - 12a-2z4 + 19a-2z6 - 9a-2z8 + 2a-1z-1 - 2a-1z - 17a-1z3 + 23a-1z5 + a-1z7 - 7a-1z9 + 1 + 7z2 - 39z4 + 57z6 - 22z8 - 2z10 + 2az-1 - 27az3 + 32az5 + 9az7 - 15az9 + 6a2z2 - 35a2z4 + 58a2z6 - 26a2z8 - 2a2z10 + a3z-1 + a3z - 21a3z3 + 35a3z5 - 8a3z7 - 8a3z9 + a4z2 - 5a4z4 + 16a4z6 - 13a4z8 - 6a5z3 + 15a5z5 - 11a5z7 + 4a6z4 - 5a6z6 - 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                     4   j = 6                   6 1   j = 4                 11 4     j = 2               12 6       j = 0             15 11         j = -2           14 14           j = -4         10 13             j = -6       7 14               j = -8     4 10                 j = -10   1 7                   j = -12   4                     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, 2]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 2]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[14, 10, 15, 9], > X[8, 4, 9, 3], X[10, 5, 11, 6], X[18, 11, 19, 12], X[20, 14, 21, 13], > X[22, 19, 5, 20], X[12, 22, 13, 21], X[2, 16, 3, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {6, -1, 2, -5, 4, -6, 7, -10, 8, -4, 11, -2, 3, -7, > 9, -8, 10, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 5 11 17 24 27 27 -q + ----- - ---- + ---- - ---- + ---- - ------- + 23 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 > 17 q + 10 q - 5 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 2 -16 4 4 4 3 2 5 5 2 4 3 + q - --- - q + --- - --- + --- + -- - -- + -- - -- - 2 q - 2 q + 18 14 12 10 8 6 4 2 q q q q q q q q 6 8 10 12 14 > 6 q - 2 q + 2 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 2]][a, z]
Out[8]=	3 3 3 1 2 2 a a 3 z 3 z 3 3 3 5 3 -(----) + --- - --- + -- + a z - a z + -- - ---- + 5 a z - 3 a z + a z - 3 a z z z 3 a a z a 5 2 z 5 3 5 7 > ---- + 3 a z - 2 a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 2]][a, z]
Out[9]=	3 2 1 2 2 a a z 2 z 3 2 2 z 2 2 4 2 1 + ---- + --- + --- + -- - -- - --- + a z + 7 z + ---- + 6 a z + a z - 3 a z z z 3 a 2 a z a a 3 3 4 4 5 z 17 z 3 3 3 5 3 4 z 12 z > ---- - ----- - 27 a z - 21 a z - 6 a z - 39 z + -- - ----- - 3 a 4 2 a a a 5 5 2 4 4 4 6 4 10 z 23 z 5 3 5 > 35 a z - 5 a z + 4 a z + ----- + ----- + 32 a z + 35 a z + 3 a a 6 6 5 5 7 5 6 z 19 z 2 6 4 6 6 6 > 15 a z - a z + 57 z - -- + ----- + 58 a z + 16 a z - 5 a z - 4 2 a a 7 7 8 5 z z 7 3 7 5 7 8 9 z 2 8 > ---- + -- + 9 a z - 8 a z - 11 a z - 22 z - ---- - 26 a z - 3 a 2 a a 9 4 8 7 z 9 3 9 10 2 10 > 13 a z - ---- - 15 a z - 8 a z - 2 z - 2 a z a
In[10]:=	Kh[L][q, t]
Out[10]=	14 1 4 1 7 4 10 7 14 15 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 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 10 13 14 2 2 2 4 2 4 3 > ----- + ---- + ---- + 11 t + 12 q t + 6 q t + 11 q t + 4 q t + 4 2 4 2 q t q t q t 6 3 6 4 8 4 10 5 > 6 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a2

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