© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11a126 K11a128

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,4,11,3 X14,5,15,6 X20,8,21,7 X2,10,3,9 X22,11,1,12 X18,14,19,13 X8,15,9,16 X12,18,13,17 X6,20,7,19 X16,22,17,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 14 20 2 22 18 8 12 6 16

Alexander Polynomial:

t-4 - 6t-3 + 17t-2 - 28t-1 + 33 - 28t + 17t2 - 6t3 + t4

Conway Polynomial:

1 + 2z2 + z4 + 2z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{137, 4}

Jones Polynomial:

- q-1 + 4 - 8q + 14q2 - 18q3 + 22q4 - 22q5 + 19q6 - 15q7 + 9q8 - 4q9 + q10

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

- q-2 + 2 - 2q2 + 2q4 + 2q6 - 2q8 + 6q10 - 3q12 + 3q14 - q16 - 3q18 + 2q20 - 4q22 + 2q24 - q28 + q30

HOMFLY-PT Polynomial:

a-8 + 2a-8z2 + a-8z4 - 4a-6 - 8a-6z2 - 7a-6z4 - 2a-6z6 + 4a-4 + 10a-4z2 + 10a-4z4 + 5a-4z6 + a-4z8 - 2a-2z2 - 3a-2z4 - a-2z6

Kauffman Polynomial:

a-12z4 - a-11z3 + 4a-11z5 + 2a-10z2 - 7a-10z4 + 9a-10z6 - 3a-9z + 12a-9z3 - 20a-9z5 + 14a-9z7 + a-8 - 3a-8z2 + 9a-8z4 - 21a-8z6 + 14a-8z8 - 10a-7z + 32a-7z3 - 36a-7z5 + 2a-7z7 + 8a-7z9 + 4a-6 - 20a-6z2 + 54a-6z4 - 64a-6z6 + 20a-6z8 + 2a-6z10 - 10a-5z + 27a-5z3 - 8a-5z5 - 25a-5z7 + 13a-5z9 + 4a-4 - 21a-4z2 + 53a-4z4 - 48a-4z6 + 10a-4z8 + 2a-4z10 - 4a-3z + 11a-3z3 + a-3z5 - 12a-3z7 + 5a-3z9 - 6a-2z2 + 16a-2z4 - 14a-2z6 + 4a-2z8 - a-1z + 3a-1z3 - 3a-1z5 + a-1z7

V2 and V3, the type 2 and 3 Vassiliev invariants:

{2, 2}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=4 is the signature of 11127. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 21                       1 j = 19                     3   j = 17                   6 1   j = 15                 9 3     j = 13               10 6       j = 11             12 9         j = 9           10 10           j = 7         8 12             j = 5       6 10               j = 3     3 9                 j = 1   1 5                   j = -1   3                     j = -3 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]:=	Show[DrawMorseLink[Knot[11, Alternating, 127]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 127]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[14, 5, 15, 6], X[20, 8, 21, 7], > X[2, 10, 3, 9], X[22, 11, 1, 12], X[18, 14, 19, 13], X[8, 15, 9, 16], > X[12, 18, 13, 17], X[6, 20, 7, 19], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 127]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -10, 4, -8, 5, -2, 6, -9, 7, -3, 8, -11, 9, -7, 10, > -4, 11, -6]
In[5]:=	DTCode[Knot[11, Alternating, 127]]
Out[5]=	DTCode[4, 10, 14, 20, 2, 22, 18, 8, 12, 6, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 127]][t]
Out[6]=	-4 6 17 28 2 3 4 33 + t - -- + -- - -- - 28 t + 17 t - 6 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 127]][z]
Out[7]=	2 4 6 8 1 + 2 z + z + 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 127]}
In[9]:=	{KnotDet[Knot[11, Alternating, 127]], KnotSignature[Knot[11, Alternating, 127]]}
Out[9]=	{137, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 127]][q]
Out[10]=	1 2 3 4 5 6 7 8 9 10 4 - - - 8 q + 14 q - 18 q + 22 q - 22 q + 19 q - 15 q + 9 q - 4 q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 127]}
In[12]:=	A2Invariant[Knot[11, Alternating, 127]][q]
Out[12]=	-2 2 4 6 8 10 12 14 16 18 2 - q - 2 q + 2 q + 2 q - 2 q + 6 q - 3 q + 3 q - q - 3 q + 20 22 24 28 30 > 2 q - 4 q + 2 q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 127]][a, z]
Out[13]=	2 2 2 2 4 4 4 4 6 -8 4 4 2 z 8 z 10 z 2 z z 7 z 10 z 3 z 2 z a - -- + -- + ---- - ---- + ----- - ---- + -- - ---- + ----- - ---- - ---- + 6 4 8 6 4 2 8 6 4 2 6 a a a a a a a a a a a 6 6 8 5 z z z > ---- - -- + -- 4 2 4 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 127]][a, z]
Out[14]=	2 2 2 2 -8 4 4 3 z 10 z 10 z 4 z z 2 z 3 z 20 z 21 z a + -- + -- - --- - ---- - ---- - --- - - + ---- - ---- - ----- - ----- - 6 4 9 7 5 3 a 10 8 6 4 a a a a a a a a a a 2 3 3 3 3 3 3 4 4 4 6 z z 12 z 32 z 27 z 11 z 3 z z 7 z 9 z > ---- - --- + ----- + ----- + ----- + ----- + ---- + --- - ---- + ---- + 2 11 9 7 5 3 a 12 10 8 a a a a a a a a a 4 4 4 5 5 5 5 5 5 6 54 z 53 z 16 z 4 z 20 z 36 z 8 z z 3 z 9 z > ----- + ----- + ----- + ---- - ----- - ----- - ---- + -- - ---- + ---- - 6 4 2 11 9 7 5 3 a 10 a a a a a a a a a 6 6 6 6 7 7 7 7 7 8 21 z 64 z 48 z 14 z 14 z 2 z 25 z 12 z z 14 z > ----- - ----- - ----- - ----- + ----- + ---- - ----- - ----- + -- + ----- + 8 6 4 2 9 7 5 3 a 8 a a a a a a a a a 8 8 8 9 9 9 10 10 20 z 10 z 4 z 8 z 13 z 5 z 2 z 2 z > ----- + ----- + ---- + ---- + ----- + ---- + ----- + ----- 6 4 2 7 5 3 6 4 a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 127]], Vassiliev[3][Knot[11, Alternating, 127]]}
Out[15]=	{2, 2}
In[16]:=	Kh[Knot[11, Alternating, 127]][q, t]
Out[16]=	3 3 5 1 3 q 5 q 3 q 5 7 7 2 9 q + 6 q + ----- + ---- + -- + --- + ---- + 10 q t + 8 q t + 12 q t + 3 3 2 2 t t q t q t t 9 2 9 3 11 3 11 4 13 4 13 5 > 10 q t + 10 q t + 12 q t + 9 q t + 10 q t + 6 q t + 15 5 15 6 17 6 17 7 19 7 21 8 > 9 q t + 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a127

K11a126 K11a128