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

K11a125 K11a127

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 X16,11,17,12 X22,13,1,14 X8,15,9,16 X12,17,13,18 X6,20,7,19 X18,22,19,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 14 20 2 16 22 8 12 6 18

Alexander Polynomial:

t-4 - 5t-3 + 16t-2 - 31t-1 + 39 - 31t + 16t2 - 5t3 + t4

Conway Polynomial:

1 + 4z2 + 6z4 + 3z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{145, 0}

Jones Polynomial:

- q-5 + 3q-4 - 8q-3 + 14q-2 - 19q-1 + 24 - 23q + 21q2 - 16q3 + 10q4 - 5q5 + q6

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

{...}

A2 (sl(3)) Invariant:

- q-14 + q-12 - 4q-10 + q-8 + q-6 - 3q-4 + 7q-2 - 1 + 5q2 + q4 - 2q6 + 3q8 - 5q10 + q12 - 2q16 + q18

HOMFLY-PT Polynomial:

a-4z2 + a-4z4 - 4a-2 - 9a-2z2 - 7a-2z4 - 2a-2z6 + 9 + 19z2 + 16z4 + 6z6 + z8 - 4a2 - 7a2z2 - 4a2z4 - a2z6

Kauffman Polynomial:

- a-6z4 + a-6z6 + a-5z + 4a-5z3 - 10a-5z5 + 5a-5z7 - 2a-4z2 + 11a-4z4 - 20a-4z6 + 9a-4z8 - 3a-3z + 15a-3z3 - 17a-3z5 - 5a-3z7 + 7a-3z9 + 4a-2 - 17a-2z2 + 41a-2z4 - 50a-2z6 + 17a-2z8 + 2a-2z10 - 10a-1z + 25a-1z3 - 13a-1z5 - 16a-1z7 + 13a-1z9 + 9 - 27z2 + 48z4 - 45z6 + 16z8 + 2z10 - 10az + 23az3 - 16az5 + 6az9 + 4a2 - 11a2z2 + 15a2z4 - 13a2z6 + 8a2z8 - 3a3z + 7a3z3 - 9a3z5 + 6a3z7 + a4z2 - 4a4z4 + 3a4z6 + a5z - 2a5z3 + a5z5

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

{4, 0}

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=0 is the signature of 11126. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                       1 j = 11                     4   j = 9                   6 1   j = 7                 10 4     j = 5               11 6       j = 3             12 10         j = 1           12 11           j = -1         8 13             j = -3       6 11               j = -5     2 8                 j = -7   1 6                   j = -9   2                     j = -11 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, 126]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 126]]
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[16, 11, 17, 12], X[22, 13, 1, 14], X[8, 15, 9, 16], > X[12, 17, 13, 18], X[6, 20, 7, 19], X[18, 22, 19, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 126]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -10, 4, -8, 5, -2, 6, -9, 7, -3, 8, -6, 9, -11, 10, > -4, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 126]]
Out[5]=	DTCode[4, 10, 14, 20, 2, 16, 22, 8, 12, 6, 18]
In[6]:=	alex = Alexander[Knot[11, Alternating, 126]][t]
Out[6]=	-4 5 16 31 2 3 4 39 + t - -- + -- - -- - 31 t + 16 t - 5 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 126]][z]
Out[7]=	2 4 6 8 1 + 4 z + 6 z + 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 126]}
In[9]:=	{KnotDet[Knot[11, Alternating, 126]], KnotSignature[Knot[11, Alternating, 126]]}
Out[9]=	{145, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 126]][q]
Out[10]=	-5 3 8 14 19 2 3 4 5 6 24 - q + -- - -- + -- - -- - 23 q + 21 q - 16 q + 10 q - 5 q + q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 126]}
In[12]:=	A2Invariant[Knot[11, Alternating, 126]][q]
Out[12]=	-14 -12 4 -8 -6 3 7 2 4 6 8 -1 - q + q - --- + q + q - -- + -- + 5 q + q - 2 q + 3 q - 10 4 2 q q q 10 12 16 18 > 5 q + q - 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 126]][a, z]
Out[13]=	2 2 4 4 4 2 2 z 9 z 2 2 4 z 7 z 2 4 9 - -- - 4 a + 19 z + -- - ---- - 7 a z + 16 z + -- - ---- - 4 a z + 2 4 2 4 2 a a a a a 6 6 2 z 2 6 8 > 6 z - ---- - a z + z 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 126]][a, z]
Out[14]=	2 4 2 z 3 z 10 z 3 5 2 2 z 9 + -- + 4 a + -- - --- - ---- - 10 a z - 3 a z + a z - 27 z - ---- - 2 5 3 a 4 a a a a 2 3 3 3 17 z 2 2 4 2 4 z 15 z 25 z 3 3 3 > ----- - 11 a z + a z + ---- + ----- + ----- + 23 a z + 7 a z - 2 5 3 a a a a 4 4 4 5 5 5 3 4 z 11 z 41 z 2 4 4 4 10 z 17 z > 2 a z + 48 z - -- + ----- + ----- + 15 a z - 4 a z - ----- - ----- - 6 4 2 5 3 a a a a a 5 6 6 6 13 z 5 3 5 5 5 6 z 20 z 50 z 2 6 > ----- - 16 a z - 9 a z + a z - 45 z + -- - ----- - ----- - 13 a z + a 6 4 2 a a a 7 7 7 8 8 4 6 5 z 5 z 16 z 3 7 8 9 z 17 z 2 8 > 3 a z + ---- - ---- - ----- + 6 a z + 16 z + ---- + ----- + 8 a z + 5 3 a 4 2 a a a a 9 9 10 7 z 13 z 9 10 2 z > ---- + ----- + 6 a z + 2 z + ----- 3 a 2 a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 126]], Vassiliev[3][Knot[11, Alternating, 126]]}
Out[15]=	{4, 0}
In[16]:=	Kh[Knot[11, Alternating, 126]][q, t]
Out[16]=	13 1 2 1 6 2 8 6 11 -- + 12 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t 8 3 3 2 5 2 5 3 7 3 > --- + 11 q t + 12 q t + 10 q t + 11 q t + 6 q t + 10 q t + q t 7 4 9 4 9 5 11 5 13 6 > 4 q t + 6 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a126

K11a125 K11a127