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

K11n127 K11n129

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X5,19,6,18 X7,15,8,14 X9,16,10,17 X2,11,3,12 X13,7,14,6 X15,20,16,21 X17,1,18,22 X19,12,20,13 X21,9,22,8

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 -18 -14 -16 2 -6 -20 -22 -12 -8

Alexander Polynomial:

t-3 - 5t-2 + 10t-1 - 11 + 10t - 5t2 + t3

Conway Polynomial:

1 - z2 + z4 + z6

Other knots with the same Alexander/Conway Polynomial:

{922, ...}

Determinant and Signature:

{43, 2}

Jones Polynomial:

- q-4 + 3q-3 - 4q-2 + 6q-1 - 7 + 7q - 6q2 + 5q3 - 3q4 + q5

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

{...}

A2 (sl(3)) Invariant:

- q-12 + q-10 + q-6 + 2q-4 - q-2 + 1 - q2 + q6 - q8 + q10 - q12 + q16

HOMFLY-PT Polynomial:

a-4 + a-4z2 - 2a-2 - 5a-2z2 - 2a-2z4 + 2 + 5z2 + 4z4 + z6 - 2a2z2 - a2z4

Kauffman Polynomial:

a-6z2 - a-5z + 3a-5z3 + a-4 - 2a-4z2 + a-4z4 + a-4z6 - 2a-3z + 7a-3z3 - 8a-3z5 + 3a-3z7 + 2a-2 - 12a-2z2 + 19a-2z4 - 15a-2z6 + 4a-2z8 - 3a-1z + 8a-1z3 - 5a-1z5 - 4a-1z7 + 2a-1z9 + 2 - 15z2 + 36z4 - 30z6 + 7z8 - 3az + 8az3 - az5 - 6az7 + 2az9 - 6a2z2 + 18a2z4 - 14a2z6 + 3a2z8 - a3z + 4a3z3 - 4a3z5 + a3z7

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

{-1, -1}

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=2 is the signature of 11128. 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 j = 11                   1 j = 9                 2   j = 7               3 1   j = 5             3 2     j = 3           4 3       j = 1         4 4         j = -1       2 3           j = -3     2 4             j = -5   1 2               j = -7   2                 j = -9 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, NonAlternating, 128]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 128]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[5, 19, 6, 18], X[7, 15, 8, 14], > X[9, 16, 10, 17], X[2, 11, 3, 12], X[13, 7, 14, 6], X[15, 20, 16, 21], > X[17, 1, 18, 22], X[19, 12, 20, 13], X[21, 9, 22, 8]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 128]]
Out[4]=	GaussCode[1, -6, 2, -1, -3, 7, -4, 11, -5, -2, 6, 10, -7, 4, -8, 5, -9, 3, -10, > 8, -11, 9]
In[5]:=	DTCode[Knot[11, NonAlternating, 128]]
Out[5]=	DTCode[4, 10, -18, -14, -16, 2, -6, -20, -22, -12, -8]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 128]][t]
Out[6]=	-3 5 10 2 3 -11 + t - -- + -- + 10 t - 5 t + t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 128]][z]
Out[7]=	2 4 6 1 - z + z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 22], Knot[11, NonAlternating, 128]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 128]], KnotSignature[Knot[11, NonAlternating, 128]]}
Out[9]=	{43, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 128]][q]
Out[10]=	-4 3 4 6 2 3 4 5 -7 - q + -- - -- + - + 7 q - 6 q + 5 q - 3 q + q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 128]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 128]][q]
Out[12]=	-12 -10 -6 2 -2 2 6 8 10 12 16 1 - q + q + q + -- - q - q + q - q + q - q + q 4 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 128]][a, z]
Out[13]=	2 2 4 -4 2 2 z 5 z 2 2 4 2 z 2 4 6 2 + a - -- + 5 z + -- - ---- - 2 a z + 4 z - ---- - a z + z 2 4 2 2 a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 128]][a, z]
Out[14]=	2 2 2 -4 2 z 2 z 3 z 3 2 z 2 z 12 z 2 + a + -- - -- - --- - --- - 3 a z - a z - 15 z + -- - ---- - ----- - 2 5 3 a 6 4 2 a a a a a a 3 3 3 4 4 2 2 3 z 7 z 8 z 3 3 3 4 z 19 z > 6 a z + ---- + ---- + ---- + 8 a z + 4 a z + 36 z + -- + ----- + 5 3 a 4 2 a a a a 5 5 6 6 2 4 8 z 5 z 5 3 5 6 z 15 z 2 6 > 18 a z - ---- - ---- - a z - 4 a z - 30 z + -- - ----- - 14 a z + 3 a 4 2 a a a 7 7 8 9 3 z 4 z 7 3 7 8 4 z 2 8 2 z 9 > ---- - ---- - 6 a z + a z + 7 z + ---- + 3 a z + ---- + 2 a z 3 a 2 a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 128]], Vassiliev[3][Knot[11, NonAlternating, 128]]}
Out[15]=	{-1, -1}
In[16]:=	Kh[Knot[11, NonAlternating, 128]][q, t]
Out[16]=	3 1 2 1 2 2 4 2 3 4 q 4 q + 4 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + --- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t q t q t 3 5 5 2 7 2 7 3 9 3 11 4 > 3 q t + 3 q t + 2 q t + 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n128

K11n127 K11n129