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DrawMorseLink

PD Presentation:

X6172 X5,12,6,13 X3849 X2,16,3,15 X16,7,17,8 X19,22,20,15 X21,14,22,11 X13,20,14,21 X9,18,10,19 X11,10,12,5 X17,1,18,4

Gauss Code:

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

Jones Polynomial:

q-21/2 - 3q-19/2 + 4q-17/2 - 7q-15/2 + 7q-13/2 - 9q-11/2 + 6q-9/2 - 7q-7/2 + 2q-5/2 - 2q-3/2

A2 (sl(3)) Invariant:

- q-32 + q-30 + q-28 + 2q-26 + 7q-24 + 6q-22 + 9q-20 + 10q-18 + 10q-16 + 12q-14 + 8q-12 + 8q-10 + 5q-8 + q-6 + 2q-4

HOMFLY-PT Polynomial:

- a3z-3 - 4a3z-1 - 5a3z - 2a3z3 + 3a5z-3 + 9a5z-1 + 7a5z + 3a5z3 + a5z5 - 3a7z-3 - 6a7z-1 - a7z + 2a7z3 + a7z5 + a9z-3 + a9z-1 - a9z - a9z3

Kauffman Polynomial:

a3z-3 - 5a3z-1 + 7a3z - 3a3z3 - 3a4z-2 + 10a4 - 8a4z2 + 2a4z4 - a4z6 + 3a5z-3 - 12a5z-1 + 19a5z - 16a5z3 + 6a5z5 - 2a5z7 - 6a6z-2 + 19a6 - 15a6z2 - a6z4 + 4a6z6 - 2a6z8 + 3a7z-3 - 12a7z-1 + 25a7z - 31a7z3 + 16a7z5 - 2a7z7 - a7z9 - 3a8z-2 + 10a8 - 7a8z2 - 10a8z4 + 15a8z6 - 5a8z8 + a9z-3 - 5a9z-1 + 16a9z - 28a9z3 + 21a9z5 - 3a9z7 - a9z9 - a10z2 - 4a10z4 + 9a10z6 - 3a10z8 + 3a11z - 10a11z3 + 11a11z5 - 3a11z7 - a12z2 + 3a12z4 - a12z6

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -2                   2 j = -4                 2 2 j = -6               5     j = -8             3 4     j = -10           6 3       j = -12         3 5         j = -14       4 4           j = -16     2 5             j = -18   1 2               j = -20   2                 j = -22 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]=	4
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 454]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 454]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 12, 6, 13], X[3, 8, 4, 9], X[2, 16, 3, 15], > X[16, 7, 17, 8], X[19, 22, 20, 15], X[21, 14, 22, 11], X[13, 20, 14, 21], > X[9, 18, 10, 19], X[11, 10, 12, 5], X[17, 1, 18, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, -3, 11}, {-10, 2, -8, 7}, {-2, -1, 5, 3, -9, 10}, > {4, -5, -11, 9, -6, 8, -7, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 3 4 7 7 9 6 7 2 2 q - ----- + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 -30 -28 2 7 6 9 10 10 12 8 8 -q + q + q + --- + --- + --- + --- + --- + --- + --- + --- + --- + 26 24 22 20 18 16 14 12 10 q q q q q q q q q 5 -6 2 > -- + q + -- 8 4 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 454]][a, z]
Out[8]=	3 5 7 9 3 5 7 9 a 3 a 3 a a 4 a 9 a 6 a a 3 5 7 -(--) + ---- - ---- + -- - ---- + ---- - ---- + -- - 5 a z + 7 a z - a z - 3 3 3 3 z z z z z z z z 9 3 3 5 3 7 3 9 3 5 5 7 5 > a z - 2 a z + 3 a z + 2 a z - a z + a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 454]][a, z]
Out[9]=	3 5 7 9 4 6 8 3 4 6 8 a 3 a 3 a a 3 a 6 a 3 a 5 a 10 a + 19 a + 10 a + -- + ---- + ---- + -- - ---- - ---- - ---- - ---- - 3 3 3 3 2 2 2 z z z z z z z z 5 7 9 12 a 12 a 5 a 3 5 7 9 11 > ----- - ----- - ---- + 7 a z + 19 a z + 25 a z + 16 a z + 3 a z - z z z 4 2 6 2 8 2 10 2 12 2 3 3 5 3 > 8 a z - 15 a z - 7 a z - a z - a z - 3 a z - 16 a z - 7 3 9 3 11 3 4 4 6 4 8 4 10 4 > 31 a z - 28 a z - 10 a z + 2 a z - a z - 10 a z - 4 a z + 12 4 5 5 7 5 9 5 11 5 4 6 6 6 > 3 a z + 6 a z + 16 a z + 21 a z + 11 a z - a z + 4 a z + 8 6 10 6 12 6 5 7 7 7 9 7 11 7 > 15 a z + 9 a z - a z - 2 a z - 2 a z - 3 a z - 3 a z - 6 8 8 8 10 8 7 9 9 9 > 2 a z - 5 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 2 1 2 2 5 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 22 9 20 8 18 8 18 7 16 7 16 6 14 6 q q q t q t q t q t q t q t q t 4 3 5 6 3 3 4 5 2 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- 14 5 12 5 12 4 10 4 10 3 8 3 8 2 6 2 4 q t q t q t q t q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n454

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