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

K11n23 K11n25

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X12,5,13,6 X2837 X14,9,15,10 X11,18,12,19 X6,13,7,14 X15,21,16,20 X17,1,18,22 X19,10,20,11 X21,17,22,16

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

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

Conway Polynomial:

1 + 2z2 + 3z4 + z6

Other knots with the same Alexander/Conway Polynomial:

{87, ...}

Determinant and Signature:

{23, 2}

Jones Polynomial:

- q-4 + 2q-3 - 3q-2 + 4q-1 - 3 + 4q - 3q2 + 2q3 - q4

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

{...}

A2 (sl(3)) Invariant:

- q-12 - q-8 + q-4 + q-2 + 3 + q2 + q4 - q8 - q12

HOMFLY-PT Polynomial:

- 2a-2 - 3a-2z2 - a-2z4 + 5 + 8z2 + 5z4 + z6 - 2a2 - 3a2z2 - a2z4

Kauffman Polynomial:

- 3a-3z + 7a-3z3 - 5a-3z5 + a-3z7 + 2a-2 - 8a-2z2 + 14a-2z4 - 10a-2z6 + 2a-2z8 - 7a-1z + 16a-1z3 - 8a-1z5 - 2a-1z7 + a-1z9 + 5 - 16z2 + 28z4 - 20z6 + 4z8 - 7az + 16az3 - 8az5 - 2az7 + az9 + 2a2 - 8a2z2 + 14a2z4 - 10a2z6 + 2a2z8 - 3a3z + 7a3z3 - 5a3z5 + a3z7

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

{2, 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=2 is the signature of 1124. 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 j = 9                 1 j = 7               1   j = 5             2 1   j = 3           2 1     j = 1         2 3       j = -1       2 2 1       j = -3     1 2           j = -5   1 2             j = -7   1               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, 24]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 24]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[12, 5, 13, 6], X[2, 8, 3, 7], > X[14, 9, 15, 10], X[11, 18, 12, 19], X[6, 13, 7, 14], X[15, 21, 16, 20], > X[17, 1, 18, 22], X[19, 10, 20, 11], X[21, 17, 22, 16]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 24]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -7, 4, -2, 5, 10, -6, -3, 7, -5, -8, 11, -9, 6, -10, > 8, -11, 9]
In[5]:=	DTCode[Knot[11, NonAlternating, 24]]
Out[5]=	DTCode[4, 8, 12, 2, 14, -18, 6, -20, -22, -10, -16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 24]][t]
Out[6]=	-3 3 5 2 3 -5 + t - -- + - + 5 t - 3 t + t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 24]][z]
Out[7]=	2 4 6 1 + 2 z + 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 7], Knot[11, NonAlternating, 24]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 24]], KnotSignature[Knot[11, NonAlternating, 24]]}
Out[9]=	{23, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 24]][q]
Out[10]=	-4 2 3 4 2 3 4 -3 - q + -- - -- + - + 4 q - 3 q + 2 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, 24]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 24]][q]
Out[12]=	-12 -8 -4 -2 2 4 8 12 3 - q - q + q + q + q + q - q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 24]][a, z]
Out[13]=	2 4 2 2 2 3 z 2 2 4 z 2 4 6 5 - -- - 2 a + 8 z - ---- - 3 a z + 5 z - -- - a z + z 2 2 2 a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 24]][a, z]
Out[14]=	2 3 2 2 3 z 7 z 3 2 8 z 2 2 7 z 5 + -- + 2 a - --- - --- - 7 a z - 3 a z - 16 z - ---- - 8 a z + ---- + 2 3 a 2 3 a a a a 3 4 5 5 16 z 3 3 3 4 14 z 2 4 5 z 8 z > ----- + 16 a z + 7 a z + 28 z + ----- + 14 a z - ---- - ---- - a 2 3 a a a 6 7 7 5 3 5 6 10 z 2 6 z 2 z 7 3 7 > 8 a z - 5 a z - 20 z - ----- - 10 a z + -- - ---- - 2 a z + a z + 2 3 a a a 8 9 8 2 z 2 8 z 9 > 4 z + ---- + 2 a z + -- + a z 2 a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 24]], Vassiliev[3][Knot[11, NonAlternating, 24]]}
Out[15]=	{2, 0}
In[16]:=	Kh[Knot[11, NonAlternating, 24]][q, t]
Out[16]=	1 3 1 1 1 2 1 2 2 2 - + 3 q + 2 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 9 5 7 4 5 4 5 3 3 3 3 2 2 q t q t q t q t q t q t q t q t 2 q 3 5 5 2 7 2 9 3 > --- + q t + 2 q t + q t + q t + q t t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n24

K11n23 K11n25