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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-7 - 3q-6 + 7q-5 - 10q-4 + 16q-3 - 16q-2 + 17q-1 - 15 + 11q - 7q2 + 4q3 - q4

A2 (sl(3)) Invariant:

q-22 + 5q-16 + 2q-14 + 4q-12 + 7q-10 + 2q-8 + 6q-6 - 4q2 + 3q4 + 2q10 - q12

HOMFLY-PT Polynomial:

a-2 - a-2z2 - a-2z4 - 2 + 2z4 + z6 + a2z-2 + 4a2 + 3a2z2 + 2a2z4 + a2z6 - 2a4z-2 - 4a4 - 4a4z2 - 2a4z4 + a6z-2 + a6 + a6z2

Kauffman Polynomial:

2a-3z3 - 3a-3z5 + a-3z7 - a-2 - 2a-2z2 + 15a-2z4 - 15a-2z6 + 4a-2z8 - 2a-1z - 2a-1z3 + 15a-1z5 - 17a-1z7 + 5a-1z9 - 1 + 22z4 - 25z6 + 3z8 + 2z10 - 6az + 4az3 + 15az5 - 27az7 + 10az9 + a2z-2 - 3a2 + 5a2z2 + 9a2z4 - 22a2z6 + 6a2z8 + 2a2z10 - 2a3z-1 + 9a3z3 - 12a3z5 - 2a3z7 + 5a3z9 + 2a4z-2 - 6a4 + 12a4z2 - 7a4z4 - 6a4z6 + 7a4z8 - 2a5z-1 + 4a5z - a5z3 - 6a5z5 + 7a5z7 + a6z-2 - 4a6 + 8a6z2 - 8a6z4 + 6a6z6 - 2a7z3 + 3a7z5 - a8z2 + a8z4

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 = 9                       1 j = 7                     3   j = 5                   4 1   j = 3                 7 3     j = 1               8 4       j = -1             9 7         j = -3           9 10           j = -5         7 7             j = -7       4 10               j = -9     3 6                 j = -11     4                   j = -13 1 3                     j = -15 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, 427]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 427]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[14, 5, 15, 6], X[20, 7, 21, 8], > X[8, 19, 9, 20], X[16, 12, 17, 11], X[10, 13, 5, 14], X[22, 18, 11, 17], > X[18, 22, 19, 21], X[2, 9, 3, 10], X[4, 16, 1, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {3, -1, 4, -5, 10, -7}, > {6, -2, 7, -3, 11, -6, 8, -9, 5, -4, 9, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-7 3 7 10 16 16 17 2 3 4 -15 + q - -- + -- - -- + -- - -- + -- + 11 q - 7 q + 4 q - q 6 5 4 3 2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 5 2 4 7 2 6 2 4 10 12 q + --- + --- + --- + --- + -- + -- - 4 q + 3 q + 2 q - q 16 14 12 10 8 6 q q q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 427]][a, z]
Out[8]=	2 4 6 2 -2 2 4 6 a 2 a a z 2 2 4 2 6 2 -2 + a + 4 a - 4 a + a + -- - ---- + -- - -- + 3 a z - 4 a z + a z + 2 2 2 2 z z z a 4 4 z 2 4 4 4 6 2 6 > 2 z - -- + 2 a z - 2 a z + z + a z 2 a
In[9]:=	Kauffman[Link[11, Alternating, 427]][a, z]
Out[9]=	2 4 6 3 5 -2 2 4 6 a 2 a a 2 a 2 a 2 z -1 - a - 3 a - 6 a - 4 a + -- + ---- + -- - ---- - ---- - --- - 6 a z + 2 2 2 z z a z z z 2 3 3 5 2 z 2 2 4 2 6 2 8 2 2 z 2 z > 4 a z - ---- + 5 a z + 12 a z + 8 a z - a z + ---- - ---- + 2 3 a a a 4 3 3 3 5 3 7 3 4 15 z 2 4 4 4 > 4 a z + 9 a z - a z - 2 a z + 22 z + ----- + 9 a z - 7 a z - 2 a 5 5 6 4 8 4 3 z 15 z 5 3 5 5 5 7 5 > 8 a z + a z - ---- + ----- + 15 a z - 12 a z - 6 a z + 3 a z - 3 a a 6 7 7 6 15 z 2 6 4 6 6 6 z 17 z 7 > 25 z - ----- - 22 a z - 6 a z + 6 a z + -- - ----- - 27 a z - 2 3 a a a 8 9 3 7 5 7 8 4 z 2 8 4 8 5 z 9 > 2 a z + 7 a z + 3 z + ---- + 6 a z + 7 a z + ---- + 10 a z + 2 a a 3 9 10 2 10 > 5 a z + 2 z + 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	10 9 1 1 3 4 3 6 4 10 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 3 q 15 6 13 6 13 5 11 4 9 4 9 3 7 3 7 2 q q t q t q t q t q t q t q t q t 7 7 9 7 t 2 3 2 3 3 5 3 > ----- + ---- + ---- + --- + 8 q t + 4 q t + 7 q t + 3 q t + 4 q t + 5 2 5 3 q q t q t q t 5 4 7 4 9 5 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a427

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