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

K11n54 K11n56

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X5,14,6,15 X2837 X16,10,17,9 X18,11,19,12 X13,6,14,7 X22,16,1,15 X20,17,21,18 X12,19,13,20 X10,22,11,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-3 + 6t-2 - 14t-1 + 19 - 14t + 6t2 - t3

Conway Polynomial:

1 + z2 - z6

Other knots with the same Alexander/Conway Polynomial:

{933, ...}

Determinant and Signature:

{61, 0}

Jones Polynomial:

- 2q-3 + 5q-2 - 7q-1 + 10 - 10q + 10q2 - 8q3 + 5q4 - 3q5 + q6

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

{...}

A2 (sl(3)) Invariant:

- 2q-10 + q-6 - q-4 + 4q-2 + 1 + 2q2 + q4 - 2q6 + q8 - 3q10 + q14 - q16 + q18

HOMFLY-PT Polynomial:

a-4 + 2a-4z2 + a-4z4 - 4a-2 - 7a-2z2 - 4a-2z4 - a-2z6 + 6 + 8z2 + 3z4 - 2a2 - 2a2z2

Kauffman Polynomial:

2a-6z2 - 3a-6z4 + a-6z6 - 2a-5z + 9a-5z3 - 10a-5z5 + 3a-5z7 + a-4 + 2a-4z4 - 7a-4z6 + 3a-4z8 - 6a-3z + 19a-3z3 - 21a-3z5 + 5a-3z7 + a-3z9 + 4a-2 - 10a-2z2 + 12a-2z4 - 14a-2z6 + 6a-2z8 - 7a-1z + 15a-1z3 - 15a-1z5 + 5a-1z7 + a-1z9 + 6 - 13z2 + 11z4 - 5z6 + 3z8 - 5az + 8az3 - 4az5 + 3az7 + 2a2 - 5a2z2 + 4a2z4 + a2z6 - 2a3z + 3a3z3

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=0 is the signature of 1155. 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 j = 13                   1 j = 11                 2   j = 9               3 1   j = 7             5 2     j = 5           5 3       j = 3         5 5         j = 1       5 5           j = -1     3 6             j = -3   2 4               j = -5   3                 j = -7 2

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, 55]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 55]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 14, 6, 15], X[2, 8, 3, 7], > X[16, 10, 17, 9], X[18, 11, 19, 12], X[13, 6, 14, 7], X[22, 16, 1, 15], > X[20, 17, 21, 18], X[12, 19, 13, 20], X[10, 22, 11, 21]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 55]]
Out[4]=	GaussCode[1, -4, 2, -1, -3, 7, 4, -2, 5, -11, 6, -10, -7, 3, 8, -5, 9, -6, 10, > -9, 11, -8]
In[5]:=	DTCode[Knot[11, NonAlternating, 55]]
Out[5]=	DTCode[4, 8, -14, 2, 16, 18, -6, 22, 20, 12, 10]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 55]][t]
Out[6]=	-3 6 14 2 3 19 - t + -- - -- - 14 t + 6 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 55]][z]
Out[7]=	2 6 1 + z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 33], Knot[11, NonAlternating, 55]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 55]], KnotSignature[Knot[11, NonAlternating, 55]]}
Out[9]=	{61, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 55]][q]
Out[10]=	2 5 7 2 3 4 5 6 10 - -- + -- - - - 10 q + 10 q - 8 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, 55]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 55]][q]
Out[12]=	2 -6 -4 4 2 4 6 8 10 14 16 18 1 - --- + q - q + -- + 2 q + q - 2 q + q - 3 q + q - q + q 10 2 q q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 55]][a, z]
Out[13]=	2 2 4 4 6 -4 4 2 2 2 z 7 z 2 2 4 z 4 z z 6 + a - -- - 2 a + 8 z + ---- - ---- - 2 a z + 3 z + -- - ---- - -- 2 4 2 4 2 2 a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 55]][a, z]
Out[14]=	2 2 -4 4 2 2 z 6 z 7 z 3 2 2 z 10 z 6 + a + -- + 2 a - --- - --- - --- - 5 a z - 2 a z - 13 z + ---- - ----- - 2 5 3 a 6 2 a a a a a 3 3 3 4 4 2 2 9 z 19 z 15 z 3 3 3 4 3 z 2 z > 5 a z + ---- + ----- + ----- + 8 a z + 3 a z + 11 z - ---- + ---- + 5 3 a 6 4 a a a a 4 5 5 5 6 6 12 z 2 4 10 z 21 z 15 z 5 6 z 7 z > ----- + 4 a z - ----- - ----- - ----- - 4 a z - 5 z + -- - ---- - 2 5 3 a 6 4 a a a a a 6 7 7 7 8 8 9 9 14 z 2 6 3 z 5 z 5 z 7 8 3 z 6 z z z > ----- + a z + ---- + ---- + ---- + 3 a z + 3 z + ---- + ---- + -- + -- 2 5 3 a 4 2 3 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 55]], Vassiliev[3][Knot[11, NonAlternating, 55]]}
Out[15]=	{1, -1}
In[16]:=	Kh[Knot[11, NonAlternating, 55]][q, t]
Out[16]=	6 2 3 2 4 3 3 3 2 - + 5 q + ----- + ----- + ----- + ---- + --- + 5 q t + 5 q t + 5 q t + q 7 3 5 2 3 2 3 q t q t q t q t q t 5 2 5 3 7 3 7 4 9 4 9 5 11 5 13 6 > 5 q t + 3 q t + 5 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 K11n55

K11n54 K11n56