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

K11n141 K11n143

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X12,3,13,4 X5,16,6,17 X7,21,8,20 X9,14,10,15 X11,19,12,18 X2,13,3,14 X15,8,16,9 X17,22,18,1 X19,11,20,10 X21,7,22,6

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

t-2 - 8t-1 + 15 - 8t + t2

Conway Polynomial:

1 - 4z2 + z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{33, 0}

Jones Polynomial:

q-6 - 2q-5 + 4q-4 - 5q-3 + 5q-2 - 6q-1 + 5 - 3q + 2q2

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

{...}

A2 (sl(3)) Invariant:

q-20 + q-18 - q-16 + q-14 - q-10 + q-8 - q-6 - q-2 - 1 + q2 - q4 + 2q6 + 2q8

HOMFLY-PT Polynomial:

2a-2 - 2 - 3z2 + a2 + a2z2 + a2z4 - a4 - 2a4z2 + a6

Kauffman Polynomial:

- 2a-2 + 2a-2z2 + a-1z + a-1z5 - 2 + 7z4 - 4z6 + z8 + 5az - 8az3 + 8az5 - 4az7 + az9 - a2 - 5a2z2 + 13a2z4 - 11a2z6 + 3a2z8 + 3a3z - 3a3z3 - 2a3z7 + a3z9 - a4 + a4z2 + 2a4z4 - 6a4z6 + 2a4z8 - a5z + 5a5z3 - 7a5z5 + 2a5z7 - a6 + 4a6z2 - 4a6z4 + a6z6

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

{-4, 3}

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 11142. 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 j = 5                 2 j = 3               1   j = 1             4 2   j = -1           3 2     j = -3         2 3       j = -5       3 3         j = -7     1 2           j = -9   1 3             j = -11   1               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, 142]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 142]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 3, 13, 4], X[5, 16, 6, 17], X[7, 21, 8, 20], > X[9, 14, 10, 15], X[11, 19, 12, 18], X[2, 13, 3, 14], X[15, 8, 16, 9], > X[17, 22, 18, 1], X[19, 11, 20, 10], X[21, 7, 22, 6]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 142]]
Out[4]=	GaussCode[1, -7, 2, -1, -3, 11, -4, 8, -5, 10, -6, -2, 7, 5, -8, 3, -9, 6, -10, > 4, -11, 9]
In[5]:=	DTCode[Knot[11, NonAlternating, 142]]
Out[5]=	DTCode[4, 12, -16, -20, -14, -18, 2, -8, -22, -10, -6]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 142]][t]
Out[6]=	-2 8 2 15 + t - - - 8 t + t t
In[7]:=	Conway[Knot[11, NonAlternating, 142]][z]
Out[7]=	2 4 1 - 4 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 142]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 142]], KnotSignature[Knot[11, NonAlternating, 142]]}
Out[9]=	{33, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 142]][q]
Out[10]=	-6 2 4 5 5 6 2 5 + q - -- + -- - -- + -- - - - 3 q + 2 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, 142]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 142]][q]
Out[12]=	-20 -18 -16 -14 -10 -8 -6 -2 2 4 6 8 -1 + q + q - q + q - q + q - q - q + q - q + 2 q + 2 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 142]][a, z]
Out[13]=	2 2 4 6 2 2 2 4 2 2 4 -2 + -- + a - a + a - 3 z + a z - 2 a z + a z 2 a
In[14]:=	Kauffman[Knot[11, NonAlternating, 142]][a, z]
Out[14]=	2 2 2 4 6 z 3 5 2 z 2 2 4 2 -2 - -- - a - a - a + - + 5 a z + 3 a z - a z + ---- - 5 a z + a z + 2 a 2 a a 6 2 3 3 3 5 3 4 2 4 4 4 > 4 a z - 8 a z - 3 a z + 5 a z + 7 z + 13 a z + 2 a z - 5 6 4 z 5 5 5 6 2 6 4 6 6 6 > 4 a z + -- + 8 a z - 7 a z - 4 z - 11 a z - 6 a z + a z - a 7 3 7 5 7 8 2 8 4 8 9 3 9 > 4 a z - 2 a z + 2 a z + z + 3 a z + 2 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 142]], Vassiliev[3][Knot[11, NonAlternating, 142]]}
Out[15]=	{-4, 3}
In[16]:=	Kh[Knot[11, NonAlternating, 142]][q, t]
Out[16]=	2 1 1 1 3 1 2 3 3 - + 4 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 2 3 3 3 5 2 > ----- + ---- + --- + 2 q t + q t + 2 q t 3 2 3 q t q t q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n142

K11n141 K11n143