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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-9/2 - 4q-7/2 + 8q-5/2 - 15q-3/2 + 19q-1/2 - 23q1/2 + 24q3/2 - 20q5/2 + 15q7/2 - 10q9/2 + 4q11/2 - q13/2

A2 (sl(3)) Invariant:

- q-14 + q-12 + 2q-10 - q-8 + 6q-6 + 4 - 5q2 + 2q4 - 5q6 + 3q10 - 2q12 + 5q14 - q18 + q20

HOMFLY-PT Polynomial:

- a-5z-1 - a-5z - a-5z3 + 2a-3z-1 + 3a-3z + 4a-3z3 + 2a-3z5 - a-1z-1 - 2a-1z - 4a-1z3 - 3a-1z5 - a-1z7 - az-1 + az + 4az3 + 2az5 + a3z-1 - a3z - a3z3

Kauffman Polynomial:

a-7z3 - a-7z5 + 4a-6z4 - 4a-6z6 - a-5z-1 + 5a-5z - 10a-5z3 + 15a-5z5 - 9a-5z7 - a-4 + 4a-4z2 - 14a-4z4 + 19a-4z6 - 11a-4z8 - 2a-3z-1 + 13a-3z - 23a-3z3 + 19a-3z5 - 7a-3z9 - 3a-2 + 6a-2z2 - 26a-2z4 + 38a-2z6 - 16a-2z8 - 2a-2z10 - a-1z-1 + 10a-1z - 18a-1z3 + 9a-1z5 + 14a-1z7 - 12a-1z9 - 2 + 2z2 - 13z4 + 27z6 - 11z8 - 2z10 + az-1 + 3az - 14az3 + 16az5 + az7 - 5az9 - a2 - a2z2 - 3a2z4 + 11a2z6 - 6a2z8 + a3z-1 + a3z - 8a3z3 + 10a3z5 - 4a3z7 - a4z2 + 2a4z4 - a4z6

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 = 14                       1 j = 12                     3   j = 10                   7 1   j = 8                 8 3     j = 6               12 7       j = 4             12 8         j = 2           11 12           j = 0         10 14             j = -2       5 9               j = -4     3 10                 j = -6   1 5                   j = -8   3                     j = -10 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, 48]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 48]]
Out[4]=	PD[X[6, 1, 7, 2], X[18, 7, 19, 8], X[20, 16, 21, 15], X[16, 20, 17, 19], > X[4, 21, 1, 22], X[12, 6, 13, 5], X[10, 4, 11, 3], X[22, 12, 5, 11], > X[14, 9, 15, 10], X[2, 14, 3, 13], X[8, 17, 9, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 7, -5}, {6, -1, 2, -11, 9, -7, 8, -6, 10, -9, 3, -4, 11, -2, > 4, -3, 5, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 4 8 15 19 3/2 5/2 q - ---- + ---- - ---- + ------- - 23 Sqrt[q] + 24 q - 20 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 13/2 > 15 q - 10 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -12 2 -8 6 2 4 6 10 12 14 4 - q + q + --- - q + -- - 5 q + 2 q - 5 q + 3 q - 2 q + 5 q - 10 6 q q 18 20 > q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 48]][a, z]
Out[8]=	3 3 3 1 2 1 a a z 3 z 2 z 3 z 4 z -(----) + ---- - --- - - + -- - -- + --- - --- + a z - a z - -- + ---- - 5 3 a z z z 5 3 a 5 3 a z a z a a a a 3 5 5 7 4 z 3 3 3 2 z 3 z 5 z > ---- + 4 a z - a z + ---- - ---- + 2 a z - -- a 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 48]][a, z]
Out[9]=	3 -4 3 2 1 2 1 a a 5 z 13 z 10 z -2 - a - -- - a - ---- - ---- - --- + - + -- + --- + ---- + ---- + 3 a z + 2 5 3 a z z z 5 3 a a a z a z a a 2 2 3 3 3 3 3 2 4 z 6 z 2 2 4 2 z 10 z 23 z 18 z > a z + 2 z + ---- + ---- - a z - a z + -- - ----- - ----- - ----- - 4 2 7 5 3 a a a a a a 4 4 4 5 3 3 3 4 4 z 14 z 26 z 2 4 4 4 z > 14 a z - 8 a z - 13 z + ---- - ----- - ----- - 3 a z + 2 a z - -- + 6 4 2 7 a a a a 5 5 5 6 6 6 15 z 19 z 9 z 5 3 5 6 4 z 19 z 38 z > ----- + ----- + ---- + 16 a z + 10 a z + 27 z - ---- + ----- + ----- + 5 3 a 6 4 2 a a a a a 7 7 8 8 2 6 4 6 9 z 14 z 7 3 7 8 11 z 16 z > 11 a z - a z - ---- + ----- + a z - 4 a z - 11 z - ----- - ----- - 5 a 4 2 a a a 9 9 10 2 8 7 z 12 z 9 10 2 z > 6 a z - ---- - ----- - 5 a z - 2 z - ----- 3 a 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 3 1 5 3 10 5 10 14 + 11 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + -- + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t q t q t q t q t q t q t q t 9 2 4 4 2 6 2 6 3 8 3 > ---- + 12 q t + 12 q t + 8 q t + 12 q t + 7 q t + 8 q t + 2 q t 8 4 10 4 10 5 12 5 14 6 > 3 q t + 7 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a48

L11a47 L11a49