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

K11a127 K11a129

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 14 20 16 2 6 12 22 8 18

Alexander Polynomial:

- t-3 + 9t-2 - 31t-1 + 47 - 31t + 9t2 - t3

Conway Polynomial:

1 - 4z2 + 3z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{129, 0}

Jones Polynomial:

q-6 - 3q-5 + 8q-4 - 13q-3 + 17q-2 - 21q-1 + 21 - 18q + 14q2 - 8q3 + 4q4 - q5

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

{...}

A2 (sl(3)) Invariant:

q-20 + q-18 - 2q-16 + 2q-14 + 2q-12 - 4q-10 + 3q-8 - 2q-6 - 2q-4 + 2q-2 - 3 + 4q2 - 3q4 + q6 + 4q8 - 3q10 + 2q12 + q14 - q16

HOMFLY-PT Polynomial:

- a-4z2 + 2a-2 + 2a-2z2 + 2a-2z4 - 2 - 5z2 - 2z4 - z6 + a2 + 3a2z2 + 3a2z4 - a4 - 3a4z2 + a6

Kauffman Polynomial:

- a-5z3 + a-5z5 + 3a-4z2 - 6a-4z4 + 4a-4z6 - a-3z + 5a-3z3 - 10a-3z5 + 7a-3z7 - 2a-2 + 7a-2z2 - 7a-2z4 - 4a-2z6 + 7a-2z8 - 2a-1z + 16a-1z3 - 26a-1z5 + 9a-1z7 + 4a-1z9 - 2 + 2z2 + 12z4 - 28z6 + 15z8 + z10 + 2az + 10az3 - 18az5 - 2az7 + 8az9 - a2 - 5a2z2 + 21a2z4 - 31a2z6 + 13a2z8 + a2z10 + 2a3z + 5a3z3 - 10a3z5 - a3z7 + 4a3z9 - a4 + 5a4z4 - 10a4z6 + 5a4z8 - a5z + 5a5z3 - 7a5z5 + 3a5z7 - a6 + 3a6z2 - 3a6z4 + a6z6

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

{-4, 3}

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 11128. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 11                       1 j = 9                     3   j = 7                   5 1   j = 5                 9 3     j = 3               9 5       j = 1             12 9         j = -1           10 10           j = -3         7 11             j = -5       6 10               j = -7     2 7                 j = -9   1 6                   j = -11   2                     j = -13 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, 128]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 128]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[14, 5, 15, 6], X[20, 8, 21, 7], > X[16, 10, 17, 9], X[2, 11, 3, 12], X[6, 13, 7, 14], X[12, 16, 13, 15], > X[22, 17, 1, 18], X[8, 20, 9, 19], X[18, 21, 19, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 128]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -7, 4, -10, 5, -2, 6, -8, 7, -3, 8, -5, 9, -11, 10, > -4, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 128]]
Out[5]=	DTCode[4, 10, 14, 20, 16, 2, 6, 12, 22, 8, 18]
In[6]:=	alex = Alexander[Knot[11, Alternating, 128]][t]
Out[6]=	-3 9 31 2 3 47 - t + -- - -- - 31 t + 9 t - t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 128]][z]
Out[7]=	2 4 6 1 - 4 z + 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 128]}
In[9]:=	{KnotDet[Knot[11, Alternating, 128]], KnotSignature[Knot[11, Alternating, 128]]}
Out[9]=	{129, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 128]][q]
Out[10]=	-6 3 8 13 17 21 2 3 4 5 21 + q - -- + -- - -- + -- - -- - 18 q + 14 q - 8 q + 4 q - q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 128]}
In[12]:=	A2Invariant[Knot[11, Alternating, 128]][q]
Out[12]=	-20 -18 2 2 2 4 3 2 2 2 2 4 -3 + q + q - --- + --- + --- - --- + -- - -- - -- + -- + 4 q - 3 q + 16 14 12 10 8 6 4 2 q q q q q q q q 6 8 10 12 14 16 > q + 4 q - 3 q + 2 q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 128]][a, z]
Out[13]=	2 2 4 2 2 4 6 2 z 2 z 2 2 4 2 4 2 z -2 + -- + a - a + a - 5 z - -- + ---- + 3 a z - 3 a z - 2 z + ---- + 2 4 2 2 a a a a 2 4 6 > 3 a z - z
In[14]:=	Kauffman[Knot[11, Alternating, 128]][a, z]
Out[14]=	2 2 2 4 6 z 2 z 3 5 2 3 z -2 - -- - a - a - a - -- - --- + 2 a z + 2 a z - a z + 2 z + ---- + 2 3 a 4 a a a 2 3 3 3 7 z 2 2 6 2 z 5 z 16 z 3 3 3 > ---- - 5 a z + 3 a z - -- + ---- + ----- + 10 a z + 5 a z + 2 5 3 a a a a 4 4 5 5 5 3 4 6 z 7 z 2 4 4 4 6 4 z 10 z > 5 a z + 12 z - ---- - ---- + 21 a z + 5 a z - 3 a z + -- - ----- - 4 2 5 3 a a a a 5 6 6 26 z 5 3 5 5 5 6 4 z 4 z 2 6 > ----- - 18 a z - 10 a z - 7 a z - 28 z + ---- - ---- - 31 a z - a 4 2 a a 7 7 8 4 6 6 6 7 z 9 z 7 3 7 5 7 8 7 z > 10 a z + a z + ---- + ---- - 2 a z - a z + 3 a z + 15 z + ---- + 3 a 2 a a 9 2 8 4 8 4 z 9 3 9 10 2 10 > 13 a z + 5 a z + ---- + 8 a z + 4 a z + z + a z a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 128]], Vassiliev[3][Knot[11, Alternating, 128]]}
Out[15]=	{-4, 3}
In[16]:=	Kh[Knot[11, Alternating, 128]][q, t]
Out[16]=	10 1 2 1 6 2 7 6 10 -- + 12 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 7 11 10 3 3 2 5 2 5 3 > ----- + ---- + --- + 9 q t + 9 q t + 5 q t + 9 q t + 3 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a128

K11a127 K11a129