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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X13,20,14,21 X16,12,17,11 X19,12,20,13 X8,16,5,15 X14,8,15,7 X17,19,18,22 X21,9,22,18 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

- q-9/2 - 4q-5/2 + 3q-3/2 - 7q-1/2 + 4q1/2 - 5q3/2 + 4q5/2 - 3q7/2 + q9/2

A2 (sl(3)) Invariant:

q-16 + 3q-14 + 5q-12 + 9q-10 + 13q-8 + 12q-6 + 13q-4 + 10q-2 + 7 + 5q2 + q4 + 2q6 + q12 - q14

HOMFLY-PT Polynomial:

a-3z + a-3z3 - a-1z-3 - 2a-1z-1 - 3a-1z - 3a-1z3 - a-1z5 + 3az-3 + 5az-1 + 4az + 2az3 - 3a3z-3 - 4a3z-1 - 2a3z + a5z-3 + a5z-1

Kauffman Polynomial:

- a-4z2 + 3a-4z4 - a-4z6 + 3a-3z - 8a-3z3 + 11a-3z5 - 3a-3z7 - a-2z2 - 7a-2z4 + 11a-2z6 - 3a-2z8 + a-1z-3 - 3a-1z-1 + 8a-1z - 19a-1z3 + 13a-1z5 - a-1z9 - 3z-2 + 6 + 2z2 - 19z4 + 17z6 - 4z8 + 3az-3 - 6az-1 + 7az - 10az3 + az5 + 3az7 - az9 - 6a2z-2 + 11a2 - a2z2 - 9a2z4 + 5a2z6 - a2z8 + 3a3z-3 - 6a3z-1 + 5a3z - a3z5 - 3a4z-2 + 6a4 - 3a4z2 + a5z-3 - 3a5z-1 + 3a5z - a5z3

Khovanov Homology:

trqj 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                 2   j = 6               2 1   j = 4             3 2     j = 2         2 3 2       j = 0       1 7 3         j = -2       3 7           j = -4     2 1             j = -6 1   3               j = -8 2 1                 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]=	4
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 448]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 448]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[13, 20, 14, 21], X[16, 12, 17, 11], > X[19, 12, 20, 13], X[8, 16, 5, 15], X[14, 8, 15, 7], X[17, 19, 18, 22], > X[21, 9, 22, 18], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 7, -6}, {-5, 3, -9, 8}, > {11, -2, 4, 5, -3, -7, 6, -4, -8, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 4 3 7 3/2 5/2 7/2 9/2 -q - ---- + ---- - ------- + 4 Sqrt[q] - 5 q + 4 q - 3 q + q 5/2 3/2 Sqrt[q] q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 3 5 9 13 12 13 10 2 4 6 12 14 7 + q + --- + --- + --- + -- + -- + -- + -- + 5 q + q + 2 q + q - q 14 12 10 8 6 4 2 q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 448]][a, z]
Out[8]=	3 5 3 5 1 3 a 3 a a 2 5 a 4 a a z 3 z 3 -(----) + --- - ---- + -- - --- + --- - ---- + -- + -- - --- + 4 a z - 2 a z + 3 3 3 3 a z z z z 3 a a z z z z a 3 3 5 z 3 z 3 z > -- - ---- + 2 a z - -- 3 a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 448]][a, z]
Out[9]=	3 5 2 4 2 4 1 3 a 3 a a 3 6 a 3 a 3 6 a 6 + 11 a + 6 a + ---- + --- + ---- + -- - -- - ---- - ---- - --- - --- - 3 3 3 3 2 2 2 a z z a z z z z z z z 3 5 2 2 6 a 3 a 3 z 8 z 3 5 2 z z > ---- - ---- + --- + --- + 7 a z + 5 a z + 3 a z + 2 z - -- - -- - z z 3 a 4 2 a a a 3 3 4 4 2 2 4 2 8 z 19 z 3 5 3 4 3 z 7 z > a z - 3 a z - ---- - ----- - 10 a z - a z - 19 z + ---- - ---- - 3 a 4 2 a a a 5 5 6 6 2 4 11 z 13 z 5 3 5 6 z 11 z 2 6 > 9 a z + ----- + ----- + a z - a z + 17 z - -- + ----- + 5 a z - 3 a 4 2 a a a 7 8 9 3 z 7 8 3 z 2 8 z 9 > ---- + 3 a z - 4 z - ---- - a z - -- - a z 3 2 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	7 2 1 2 1 1 3 2 1 1 7 + -- + 2 q + ------ + ----- + ----- + ----- + ----- + ----- + - + ---- + 2 10 4 8 4 6 4 8 3 6 2 4 2 t 4 q q t q t q t q t q t q t q t 3 2 2 2 4 2 4 3 6 3 6 4 > ---- + 3 t + 3 q t + 2 q t + 3 q t + 2 q t + 2 q t + q t + 2 q t 8 4 10 5 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n448

L11n447 L11n449