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

K11n156 K11n158

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X3,11,4,10 X5,12,6,13 X14,7,15,8 X9,16,10,17 X11,19,12,18 X22,13,1,14 X20,16,21,15 X17,4,18,5 X19,3,20,2 X8,21,9,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-3 + 6t-2 - 15t-1 + 21 - 15t + 6t2 - t3

Conway Polynomial:

1 - z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{65, 0}

Jones Polynomial:

q-6 - 4q-5 + 7q-4 - 9q-3 + 11q-2 - 11q-1 + 10 - 7q + 4q2 - q3

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

{...}

A2 (sl(3)) Invariant:

q-18 - 2q-16 + q-14 - 2q-10 + 3q-8 - q-6 + 2q-4 - 1 + 2q2 - 2q4 + 2q6 + q8 - q10

HOMFLY-PT Polynomial:

- a-2z2 + 1 + 3z2 + 2z4 - 3a2z2 - 3a2z4 - a2z6 + a4z2 + a4z4

Kauffman Polynomial:

a-3z3 - 2a-2z2 + 4a-2z4 - a-1z3 + 3a-1z5 + a-1z7 + 1 - 5z2 + 10z4 - 6z6 + 3z8 + az3 - 2az5 - az7 + 2az9 - 3a2z2 + 13a2z4 - 20a2z6 + 8a2z8 + 9a3z3 - 16a3z5 + 2a3z7 + 2a3z9 + a4z2 + 5a4z4 - 13a4z6 + 5a4z8 + 6a5z3 - 11a5z5 + 4a5z7 + a6z2 - 2a6z4 + a6z6

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

{0, 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=0 is the signature of 11157. 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 j = 7                   1 j = 5                 3   j = 3               4 1   j = 1             6 3     j = -1           6 5       j = -3         5 5         j = -5       4 6           j = -7     3 5             j = -9   1 4               j = -11   3                 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, NonAlternating, 157]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 157]]
Out[3]=	PD[X[6, 2, 7, 1], X[3, 11, 4, 10], X[5, 12, 6, 13], X[14, 7, 15, 8], > X[9, 16, 10, 17], X[11, 19, 12, 18], X[22, 13, 1, 14], X[20, 16, 21, 15], > X[17, 4, 18, 5], X[19, 3, 20, 2], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 157]]
Out[4]=	GaussCode[1, 10, -2, 9, -3, -1, 4, -11, -5, 2, -6, 3, 7, -4, 8, 5, -9, 6, -10, > -8, 11, -7]
In[5]:=	DTCode[Knot[11, NonAlternating, 157]]
Out[5]=	DTCode[6, -10, -12, 14, -16, -18, 22, 20, -4, -2, 8]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 157]][t]
Out[6]=	-3 6 15 2 3 21 - t + -- - -- - 15 t + 6 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 157]][z]
Out[7]=	6 1 - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 157]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 157]], KnotSignature[Knot[11, NonAlternating, 157]]}
Out[9]=	{65, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 157]][q]
Out[10]=	-6 4 7 9 11 11 2 3 10 + q - -- + -- - -- + -- - -- - 7 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, NonAlternating, 157]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 157]][q]
Out[12]=	-18 2 -14 2 3 -6 2 2 4 6 8 10 -1 + q - --- + q - --- + -- - q + -- + 2 q - 2 q + 2 q + q - q 16 10 8 4 q q q q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 157]][a, z]
Out[13]=	2 2 z 2 2 4 2 4 2 4 4 4 2 6 1 + 3 z - -- - 3 a z + a z + 2 z - 3 a z + a z - a z 2 a
In[14]:=	Kauffman[Knot[11, NonAlternating, 157]][a, z]
Out[14]=	2 3 3 2 2 z 2 2 4 2 6 2 z z 3 3 3 1 - 5 z - ---- - 3 a z + a z + a z + -- - -- + a z + 9 a z + 2 3 a a a 4 5 5 3 4 4 z 2 4 4 4 6 4 3 z 5 > 6 a z + 10 z + ---- + 13 a z + 5 a z - 2 a z + ---- - 2 a z - 2 a a 7 3 5 5 5 6 2 6 4 6 6 6 z 7 > 16 a z - 11 a z - 6 z - 20 a z - 13 a z + a z + -- - a z + a 3 7 5 7 8 2 8 4 8 9 3 9 > 2 a z + 4 a z + 3 z + 8 a z + 5 a z + 2 a z + 2 a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 157]], Vassiliev[3][Knot[11, NonAlternating, 157]]}
Out[15]=	{0, 0}
In[16]:=	Kh[Knot[11, NonAlternating, 157]][q, t]
Out[16]=	5 1 3 1 4 3 5 4 6 - + 6 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 5 5 6 3 3 2 5 2 7 3 > ----- + ---- + --- + 3 q t + 4 q t + q t + 3 q t + q t 3 2 3 q t q t q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n157

K11n156 K11n158