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

K11a142 K11a144

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X16,5,17,6 X18,7,19,8 X12,10,13,9 X2,11,3,12 X20,13,21,14 X22,15,1,16 X8,17,9,18 X6,19,7,20 X14,21,15,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 16 18 12 2 20 22 8 6 14

Alexander Polynomial:

- 2t-3 + 11t-2 - 20t-1 + 23 - 20t + 11t2 - 2t3

Conway Polynomial:

1 + 6z2 - z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{89, -4}

Jones Polynomial:

- q-11 + 2q-10 - 5q-9 + 9q-8 - 12q-7 + 14q-6 - 14q-5 + 13q-4 - 9q-3 + 6q-2 - 3q-1 + 1

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

{...}

A2 (sl(3)) Invariant:

- q-34 - q-32 - 2q-28 + 2q-26 + 2q-24 - q-22 + 2q-20 - 3q-18 + q-16 + q-14 + 4q-10 - 2q-8 + q-6 - q-2 + 1

HOMFLY-PT Polynomial:

2a2z2 + a2z4 + 2a4 + a4z2 - 2a4z4 - a4z6 - 2a6 - a6z2 - 2a6z4 - a6z6 + 3a8 + 5a8z2 + 2a8z4 - 2a10 - a10z2

Kauffman Polynomial:

2a2z2 - 3a2z4 + a2z6 + 6a3z3 - 9a3z5 + 3a3z7 + 2a4 - 6a4z2 + 9a4z4 - 11a4z6 + 4a4z8 + a5z - 3a5z3 + a5z5 - 5a5z7 + 3a5z9 + 2a6 - 14a6z2 + 25a6z4 - 20a6z6 + 5a6z8 + a6z10 + 3a7z - 13a7z3 + 21a7z5 - 16a7z7 + 6a7z9 + 3a8 - 17a8z2 + 32a8z4 - 20a8z6 + 5a8z8 + a8z10 - a9z + a9z3 + 5a9z5 - 5a9z7 + 3a9z9 + 2a10 - 10a10z2 + 15a10z4 - 10a10z6 + 4a10z8 - a11z + 2a11z3 - 5a11z5 + 3a11z7 + a12z2 - 4a12z4 + 2a12z6 + 2a13z - 3a13z3 + a13z5

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

{6, -16}

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=-4 is the signature of 11143. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 1                       1 j = -1                     2   j = -3                   4 1   j = -5                 6 3     j = -7               7 3       j = -9             7 6         j = -11           7 7           j = -13         5 7             j = -15       4 7               j = -17     1 5                 j = -19   1 4                   j = -21   1                     j = -23 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, 143]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 143]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[16, 5, 17, 6], X[18, 7, 19, 8], > X[12, 10, 13, 9], X[2, 11, 3, 12], X[20, 13, 21, 14], X[22, 15, 1, 16], > X[8, 17, 9, 18], X[6, 19, 7, 20], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 143]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -9, 5, -2, 6, -5, 7, -11, 8, -3, 9, -4, 10, > -7, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 143]]
Out[5]=	DTCode[4, 10, 16, 18, 12, 2, 20, 22, 8, 6, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 143]][t]
Out[6]=	2 11 20 2 3 23 - -- + -- - -- - 20 t + 11 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 143]][z]
Out[7]=	2 4 6 1 + 6 z - z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 143]}
In[9]:=	{KnotDet[Knot[11, Alternating, 143]], KnotSignature[Knot[11, Alternating, 143]]}
Out[9]=	{89, -4}
In[10]:=	J=Jones[Knot[11, Alternating, 143]][q]
Out[10]=	-11 2 5 9 12 14 14 13 9 6 3 1 - q + --- - -- + -- - -- + -- - -- + -- - -- + -- - - 10 9 8 7 6 5 4 3 2 q q q q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 143]}
In[12]:=	A2Invariant[Knot[11, Alternating, 143]][q]
Out[12]=	-34 -32 2 2 2 -22 2 3 -16 -14 4 2 1 - q - q - --- + --- + --- - q + --- - --- + q + q + --- - -- + 28 26 24 20 18 10 8 q q q q q q q -6 -2 > q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 143]][a, z]
Out[13]=	4 6 8 10 2 2 4 2 6 2 8 2 10 2 2 a - 2 a + 3 a - 2 a + 2 a z + a z - a z + 5 a z - a z + 2 4 4 4 6 4 8 4 4 6 6 6 > a z - 2 a z - 2 a z + 2 a z - a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 143]][a, z]
Out[14]=	4 6 8 10 5 7 9 11 13 2 2 2 a + 2 a + 3 a + 2 a + a z + 3 a z - a z - a z + 2 a z + 2 a z - 4 2 6 2 8 2 10 2 12 2 3 3 5 3 > 6 a z - 14 a z - 17 a z - 10 a z + a z + 6 a z - 3 a z - 7 3 9 3 11 3 13 3 2 4 4 4 6 4 > 13 a z + a z + 2 a z - 3 a z - 3 a z + 9 a z + 25 a z + 8 4 10 4 12 4 3 5 5 5 7 5 9 5 > 32 a z + 15 a z - 4 a z - 9 a z + a z + 21 a z + 5 a z - 11 5 13 5 2 6 4 6 6 6 8 6 10 6 > 5 a z + a z + a z - 11 a z - 20 a z - 20 a z - 10 a z + 12 6 3 7 5 7 7 7 9 7 11 7 4 8 > 2 a z + 3 a z - 5 a z - 16 a z - 5 a z + 3 a z + 4 a z + 6 8 8 8 10 8 5 9 7 9 9 9 6 10 8 10 > 5 a z + 5 a z + 4 a z + 3 a z + 6 a z + 3 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 143]], Vassiliev[3][Knot[11, Alternating, 143]]}
Out[15]=	{6, -16}
In[16]:=	Kh[Knot[11, Alternating, 143]][q, t]
Out[16]=	3 4 1 1 1 4 1 5 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 23 9 21 8 19 8 19 7 17 7 17 6 15 6 q q q t q t q t q t q t q t q t 7 5 7 7 7 7 6 7 3 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 15 5 13 5 13 4 11 4 11 3 9 3 9 2 7 2 7 q t q t q t q t q t q t q t q t q t 6 t 2 t 2 > ---- + -- + --- + q t 5 3 q q t q

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a143

K11a142 K11a144