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L11a377 L11a379

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DrawMorseLink

PD Presentation:

X12,1,13,2 X2,13,3,14 X14,3,15,4 X4,11,5,12 X16,8,17,7 X18,6,19,5 X22,16,11,15 X6,18,7,17 X8,22,9,21 X20,10,21,9 X10,20,1,19

Gauss Code:

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

Jones Polynomial:

- q-9/2 + 3q-7/2 - 6q-5/2 + 9q-3/2 - 11q-1/2 + 12q1/2 - 12q3/2 + 9q5/2 - 8q7/2 + 4q9/2 - 2q11/2 + q13/2

A2 (sl(3)) Invariant:

q-14 - q-12 + 2q-8 - 2q-6 + q-4 - q-2 - 1 + 2q2 + 4q6 + 2q8 + q10 + 3q12 - q14 - q20

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

8a-6z2 - 12a-6z4 + 6a-6z6 - a-6z8 - 6a-5z + 17a-5z3 - 21a-5z5 + 11a-5z7 - 2a-5z9 + 15a-4z2 - 31a-4z4 + 16a-4z6 - a-4z10 + a-3z-1 - 8a-3z + 14a-3z3 - 28a-3z5 + 24a-3z7 - 6a-3z9 - a-2 + 13a-2z2 - 40a-2z4 + 36a-2z6 - 7a-2z8 - a-2z10 + a-1z-1 + 2a-1z - 18a-1z3 + 19a-1z5 + 3a-1z7 - 4a-1z9 + z2 - 5z4 + 17z6 - 8z8 + 3az - 10az3 + 20az5 - 10az7 - 5a2z2 + 13a2z4 - 9a2z6 - a3z + 4a3z3 - 6a3z5 - 3a4z4 - a5z3

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 14                       1 j = 12                     1   j = 10                   3 1   j = 8                 5 1     j = 6               4 3       j = 4             8 5         j = 2           5 5           j = 0         6 7             j = -2       4 6               j = -4     2 5                 j = -6   1 4                   j = -8   2                     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, 378]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 378]]
Out[4]=	PD[X[12, 1, 13, 2], X[2, 13, 3, 14], X[14, 3, 15, 4], X[4, 11, 5, 12], > X[16, 8, 17, 7], X[18, 6, 19, 5], X[22, 16, 11, 15], X[6, 18, 7, 17], > X[8, 22, 9, 21], X[20, 10, 21, 9], X[10, 20, 1, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -4, 6, -8, 5, -9, 10, -11}, > {4, -1, 2, -3, 7, -5, 8, -6, 11, -10, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 3 6 9 11 3/2 5/2 -q + ---- - ---- + ---- - ------- + 12 Sqrt[q] - 12 q + 9 q - 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 13/2 > 8 q + 4 q - 2 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -12 2 2 -4 -2 2 6 8 10 12 -1 + q - q + -- - -- + q - q + 2 q + 4 q + 2 q + q + 3 q - 8 6 q q 14 20 > q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 378]][a, z]
Out[8]=	3 3 3 1 1 2 z z 2 z 3 z 2 z 5 z 3 3 3 -(----) + --- + --- - -- - --- + a z + a z + -- - ---- - ---- - a z + a z - 3 a z 5 3 a 5 3 a a z a a a a 5 5 z 2 z 5 > -- - ---- - a z 3 a a
In[9]:=	Kauffman[Link[11, Alternating, 378]][a, z]
Out[9]=	2 2 -2 1 1 6 z 8 z 2 z 3 2 8 z 15 z -a + ---- + --- - --- - --- + --- + 3 a z - a z + z + ---- + ----- + 3 a z 5 3 a 6 4 a z a a a a 2 3 3 3 13 z 2 2 17 z 14 z 18 z 3 3 3 5 3 > ----- - 5 a z + ----- + ----- - ----- - 10 a z + 4 a z - a z - 2 5 3 a a a a 4 4 4 5 5 5 4 12 z 31 z 40 z 2 4 4 4 21 z 28 z 19 z > 5 z - ----- - ----- - ----- + 13 a z - 3 a z - ----- - ----- + ----- + 6 4 2 5 3 a a a a a a 6 6 6 7 5 3 5 6 6 z 16 z 36 z 2 6 11 z > 20 a z - 6 a z + 17 z + ---- + ----- + ----- - 9 a z + ----- + 6 4 2 5 a a a a 7 7 8 8 9 9 9 10 10 24 z 3 z 7 8 z 7 z 2 z 6 z 4 z z z > ----- + ---- - 10 a z - 8 z - -- - ---- - ---- - ---- - ---- - --- - --- 3 a 6 2 5 3 a 4 2 a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	6 1 2 1 4 2 5 4 2 6 + -- + ------ + ----- + ----- + ----- + ----- + ---- + ---- + 7 t + 5 q t + 2 10 4 8 3 6 3 6 2 4 2 4 2 q q t q t q t q t q t q t q t 2 2 4 2 4 3 6 3 6 4 8 4 8 5 > 5 q t + 8 q t + 5 q t + 4 q t + 3 q t + 5 q t + q t + 10 5 10 6 12 6 14 7 > 3 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a378

L11a377 L11a379