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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-6 - 3q-5 + 8q-4 - 12q-3 + 19q-2 - 20q-1 + 22 - 19q + 14q2 - 9q3 + 4q4 - q5

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

- a-2z-2 - 2a-2 - 3a-2z2 - 3a-2z4 - a-2z6 + 4z-2 + 9 + 11z2 + 10z4 + 5z6 + z8 - 5a2z-2 - 10a2 - 12a2z2 - 8a2z4 - 2a2z6 + 2a4z-2 + 3a4 + 3a4z2 + a4z4

Kauffman Polynomial:

- a-5z3 + a-5z5 - 5a-4z4 + 4a-4z6 + a-3z-1 - 2a-3z + 6a-3z3 - 13a-3z5 + 8a-3z7 - a-2z-2 + 2a-2 - 8a-2z2 + 16a-2z4 - 19a-2z6 + 10a-2z8 + 5a-1z-1 - 13a-1z + 16a-1z3 - 7a-1z5 - 6a-1z7 + 7a-1z9 - 4z-2 + 13 - 30z2 + 53z4 - 45z6 + 14z8 + 2z10 + 9az-1 - 27az + 26az3 + 5az5 - 23az7 + 12az9 - 5a2z-2 + 20a2 - 46a2z2 + 59a2z4 - 41a2z6 + 10a2z8 + 2a2z10 + 5a3z-1 - 16a3z + 20a3z3 - 9a3z5 - 6a3z7 + 5a3z9 - 2a4z-2 + 10a4 - 22a4z2 + 24a4z4 - 18a4z6 + 6a4z8 + 3a5z3 - 7a5z5 + 3a5z7 + 2a6z2 - 3a6z4 + a6z6

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 = 11                       1 j = 9                     3   j = 7                   6 1   j = 5                 8 3     j = 3               11 6       j = 1             11 8         j = -1           11 13           j = -3         8 9             j = -5       5 12               j = -7     3 7                 j = -9     5                   j = -11 1 3                     j = -13 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, 428]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 428]]
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[10, 13, 5, 14], X[16, 22, 17, 21], X[18, 12, 19, 11], > X[22, 18, 11, 17], 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, -6}, > {8, -2, 6, -3, 11, -7, 9, -8, 5, -4, 7, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-6 3 8 12 19 20 2 3 4 5 22 + q - -- + -- - -- + -- - -- - 19 q + 14 q - 9 q + 4 q - q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -14 5 2 10 6 4 6 2 4 6 8 -4 + q + q + --- + --- + -- + -- + -- + -- + 3 q - 5 q - q + q - 12 10 8 6 4 2 q q q q q q 10 12 14 > 3 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 428]][a, z]
Out[8]=	2 4 2 2 2 4 4 1 5 a 2 a 2 3 z 2 2 9 - -- - 10 a + 3 a + -- - ----- - ---- + ---- + 11 z - ---- - 12 a z + 2 2 2 2 2 2 2 a z a z z z a 4 6 4 2 4 3 z 2 4 4 4 6 z 2 6 8 > 3 a z + 10 z - ---- - 8 a z + a z + 5 z - -- - 2 a z + z 2 2 a a
In[9]:=	Kauffman[Link[11, Alternating, 428]][a, z]
Out[9]=	2 4 3 2 2 4 4 1 5 a 2 a 1 5 9 a 5 a 13 + -- + 20 a + 10 a - -- - ----- - ---- - ---- + ---- + --- + --- + ---- - 2 2 2 2 2 2 3 a z z z a z a z z z a z 2 2 z 13 z 3 2 8 z 2 2 4 2 > --- - ---- - 27 a z - 16 a z - 30 z - ---- - 46 a z - 22 a z + 3 a 2 a a 3 3 3 4 6 2 z 6 z 16 z 3 3 3 5 3 4 5 z > 2 a z - -- + ---- + ----- + 26 a z + 20 a z + 3 a z + 53 z - ---- + 5 3 a 4 a a a 4 5 5 5 16 z 2 4 4 4 6 4 z 13 z 7 z 5 > ----- + 59 a z + 24 a z - 3 a z + -- - ----- - ---- + 5 a z - 2 5 3 a a a a 6 6 3 5 5 5 6 4 z 19 z 2 6 4 6 6 6 > 9 a z - 7 a z - 45 z + ---- - ----- - 41 a z - 18 a z + a z + 4 2 a a 7 7 8 8 z 6 z 7 3 7 5 7 8 10 z 2 8 > ---- - ---- - 23 a z - 6 a z + 3 a z + 14 z + ----- + 10 a z + 3 a 2 a a 9 4 8 7 z 9 3 9 10 2 10 > 6 a z + ---- + 12 a z + 5 a z + 2 z + 2 a z a
In[10]:=	Kh[L][q, t]
Out[10]=	13 1 1 3 5 3 7 5 12 -- + 11 q + ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + q 13 6 11 6 11 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 8 9 11 3 3 2 5 2 5 3 > ----- + ---- + --- + 8 q t + 11 q t + 6 q t + 8 q t + 3 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a428

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