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

K11a54 K11a56

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X16,6,17,5 X2837 X18,9,19,10 X20,11,21,12 X22,13,1,14 X6,16,7,15 X14,17,15,18 X10,19,11,20 X12,21,13,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 16 2 18 20 22 6 14 10 12

Alexander Polynomial:

- t-4 + 5t-3 - 10t-2 + 13t-1 - 13 + 13t - 10t2 + 5t3 - t4

Conway Polynomial:

1 + 2z2 - 3z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{71, -2}

Jones Polynomial:

q-7 - 3q-6 + 5q-5 - 8q-4 + 10q-3 - 11q-2 + 11q-1 - 8 + 7q - 4q2 + 2q3 - q4

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

{...}

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

- 3a-2 - 4a-2z2 - a-2z4 + 8 + 15z2 + 10z4 + 2z6 - 5a2 - 13a2z2 - 13a2z4 - 6a2z6 - a2z8 + a4 + 4a4z2 + 4a4z4 + a4z6

Kauffman Polynomial:

- 3a-3z + 7a-3z3 - 5a-3z5 + a-3z7 + 3a-2 - 8a-2z2 + 12a-2z4 - 9a-2z6 + 2a-2z8 - 6a-1z + 10a-1z3 - a-1z5 - 6a-1z7 + 2a-1z9 + 8 - 22z2 + 29z4 - 17z6 + z8 + z10 - 5az + 2az3 + 14az5 - 17az7 + 5az9 + 5a2 - 19a2z2 + 29a2z4 - 19a2z6 + 3a2z8 + a2z10 - a3z + a3z3 + 4a3z5 - 6a3z7 + 3a3z9 + a4 - 4a4z2 + 7a4z4 - 7a4z6 + 4a4z8 + 2a5z - 2a5z3 - 3a5z5 + 4a5z7 - 4a6z4 + 4a6z6 + a7z - 4a7z3 + 3a7z5 - a8z2 + a8z4

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

{2, 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 1155. 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 = 9                       1 j = 7                     1   j = 5                   3 1   j = 3                 4 1     j = 1               4 3       j = -1             7 4         j = -3           5 5           j = -5         5 6             j = -7       3 5               j = -9     2 5                 j = -11   1 3                   j = -13   2                     j = -15 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, 55]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 55]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[16, 6, 17, 5], X[2, 8, 3, 7], > X[18, 9, 19, 10], X[20, 11, 21, 12], X[22, 13, 1, 14], X[6, 16, 7, 15], > X[14, 17, 15, 18], X[10, 19, 11, 20], X[12, 21, 13, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 55]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -8, 4, -2, 5, -10, 6, -11, 7, -9, 8, -3, 9, -5, 10, > -6, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 55]]
Out[5]=	DTCode[4, 8, 16, 2, 18, 20, 22, 6, 14, 10, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 55]][t]
Out[6]=	-4 5 10 13 2 3 4 -13 - t + -- - -- + -- + 13 t - 10 t + 5 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 55]][z]
Out[7]=	2 6 8 1 + 2 z - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 55]}
In[9]:=	{KnotDet[Knot[11, Alternating, 55]], KnotSignature[Knot[11, Alternating, 55]]}
Out[9]=	{71, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 55]][q]
Out[10]=	-7 3 5 8 10 11 11 2 3 4 -8 + q - -- + -- - -- + -- - -- + -- + 7 q - 4 q + 2 q - q 6 5 4 3 2 q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 55]}
In[12]:=	A2Invariant[Knot[11, Alternating, 55]][q]
Out[12]=	-20 -18 -16 -14 -12 3 2 -4 3 2 4 6 3 + q - q + q - q - q - -- + -- - q + -- + q + 2 q - q - 8 6 2 q q q 10 12 > q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 55]][a, z]
Out[13]=	2 4 3 2 4 2 4 z 2 2 4 2 4 z 8 - -- - 5 a + a + 15 z - ---- - 13 a z + 4 a z + 10 z - -- - 2 2 2 a a a 2 4 4 4 6 2 6 4 6 2 8 > 13 a z + 4 a z + 2 z - 6 a z + a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 55]][a, z]
Out[14]=	2 3 2 4 3 z 6 z 3 5 7 2 8 z 8 + -- + 5 a + a - --- - --- - 5 a z - a z + 2 a z + a z - 22 z - ---- - 2 3 a 2 a a a 3 3 2 2 4 2 8 2 7 z 10 z 3 3 3 5 3 > 19 a z - 4 a z - a z + ---- + ----- + 2 a z + a z - 2 a z - 3 a a 4 5 7 3 4 12 z 2 4 4 4 6 4 8 4 5 z > 4 a z + 29 z + ----- + 29 a z + 7 a z - 4 a z + a z - ---- - 2 3 a a 5 6 z 5 3 5 5 5 7 5 6 9 z 2 6 > -- + 14 a z + 4 a z - 3 a z + 3 a z - 17 z - ---- - 19 a z - a 2 a 7 7 8 4 6 6 6 z 6 z 7 3 7 5 7 8 2 z > 7 a z + 4 a z + -- - ---- - 17 a z - 6 a z + 4 a z + z + ---- + 3 a 2 a a 9 2 8 4 8 2 z 9 3 9 10 2 10 > 3 a z + 4 a z + ---- + 5 a z + 3 a z + z + a z a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 55]], Vassiliev[3][Knot[11, Alternating, 55]]}
Out[15]=	{2, 1}
In[16]:=	Kh[Knot[11, Alternating, 55]][q, t]
Out[16]=	5 7 1 2 1 3 2 5 3 5 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 7 2 q q t q t q t q t q t q t q t q t 5 6 5 4 t 2 3 2 3 3 5 3 > ----- + ---- + ---- + --- + 4 q t + 3 q t + 4 q t + q t + 3 q t + 5 2 5 3 q q t q t q t 5 4 7 4 9 5 > q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a55

K11a54 K11a56