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

K11a141 K11a143

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 X6,17,7,18 X8,19,9,20 X14,21,15,22

Gauss Code:

{1, -6, 2, -1, 3, -9, 4, -10, 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 6 8 14

Alexander Polynomial:

- t-4 + 5t-3 - 8t-2 + 10t-1 - 11 + 10t - 8t2 + 5t3 - t4

Conway Polynomial:

1 + 7z2 + 2z4 - 3z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{59, -6}

Jones Polynomial:

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

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

{...}

A2 (sl(3)) Invariant:

- q-36 - q-34 - q-30 + q-28 - q-26 + q-24 + q-22 + 3q-18 - q-16 + q-14 - q-12 + q-8 + q-4

HOMFLY-PT Polynomial:

2a4 + 7a4z2 + 5a4z4 + a4z6 - 3a6 - 10a6z2 - 12a6z4 - 6a6z6 - a6z8 + 5a8 + 14a8z2 + 10a8z4 + 2a8z6 - 3a10 - 4a10z2 - a10z4

Kauffman Polynomial:

2a4 - 9a4z2 + 12a4z4 - 6a4z6 + a4z8 + a5z - 11a5z3 + 19a5z5 - 11a5z7 + 2a5z9 + 3a6 - 16a6z2 + 28a6z4 - 13a6z6 - a6z8 + a6z10 + 2a7z - 11a7z3 + 27a7z5 - 22a7z7 + 5a7z9 + 5a8 - 21a8z2 + 38a8z4 - 26a8z6 + 3a8z8 + a8z10 - 3a9z + 10a9z3 - 6a9z5 - 6a9z7 + 3a9z9 + 3a10 - 11a10z2 + 15a10z4 - 15a10z6 + 5a10z8 - 2a11z + 7a11z3 - 11a11z5 + 5a11z7 + 2a12z2 - 5a12z4 + 4a12z6 + a13z - 2a13z3 + 3a13z5 - a14z2 + 2a14z4 - a15z + a15z3

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

{7, -20}

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=-6 is the signature of 11142. 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 = -3                     1   j = -5                   3 1   j = -7                 4 2     j = -9               4 2       j = -11             4 4         j = -13           5 4           j = -15         3 4             j = -17       3 5               j = -19     1 3                 j = -21   1 3                   j = -23   1                     j = -25 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, 142]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 142]]
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[6, 17, 7, 18], X[8, 19, 9, 20], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 142]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -9, 4, -10, 5, -2, 6, -5, 7, -11, 8, -3, 9, -4, 10, > -7, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 142]]
Out[5]=	DTCode[4, 10, 16, 18, 12, 2, 20, 22, 6, 8, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 142]][t]
Out[6]=	-4 5 8 10 2 3 4 -11 - t + -- - -- + -- + 10 t - 8 t + 5 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 142]][z]
Out[7]=	2 4 6 8 1 + 7 z + 2 z - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 142]}
In[9]:=	{KnotDet[Knot[11, Alternating, 142]], KnotSignature[Knot[11, Alternating, 142]]}
Out[9]=	{59, -6}
In[10]:=	J=Jones[Knot[11, Alternating, 142]][q]
Out[10]=	-12 2 4 6 8 9 8 8 6 4 2 1 -q + --- - --- + -- - -- + -- - -- + -- - -- + -- - -- + - 11 10 9 8 7 6 5 4 3 2 q 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, 142]}
In[12]:=	A2Invariant[Knot[11, Alternating, 142]][q]
Out[12]=	-36 -34 -30 -28 -26 -24 -22 3 -16 -14 -12 -q - q - q + q - q + q + q + --- - q + q - q + 18 q -8 -4 > q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 142]][a, z]
Out[13]=	4 6 8 10 4 2 6 2 8 2 10 2 2 a - 3 a + 5 a - 3 a + 7 a z - 10 a z + 14 a z - 4 a z + 4 4 6 4 8 4 10 4 4 6 6 6 8 6 6 8 > 5 a z - 12 a z + 10 a z - a z + a z - 6 a z + 2 a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 142]][a, z]
Out[14]=	4 6 8 10 5 7 9 11 13 15 2 a + 3 a + 5 a + 3 a + a z + 2 a z - 3 a z - 2 a z + a z - a z - 4 2 6 2 8 2 10 2 12 2 14 2 5 3 > 9 a z - 16 a z - 21 a z - 11 a z + 2 a z - a z - 11 a z - 7 3 9 3 11 3 13 3 15 3 4 4 6 4 > 11 a z + 10 a z + 7 a z - 2 a z + a z + 12 a z + 28 a z + 8 4 10 4 12 4 14 4 5 5 7 5 > 38 a z + 15 a z - 5 a z + 2 a z + 19 a z + 27 a z - 9 5 11 5 13 5 4 6 6 6 8 6 > 6 a z - 11 a z + 3 a z - 6 a z - 13 a z - 26 a z - 10 6 12 6 5 7 7 7 9 7 11 7 4 8 > 15 a z + 4 a z - 11 a z - 22 a z - 6 a z + 5 a z + a z - 6 8 8 8 10 8 5 9 7 9 9 9 6 10 8 10 > a z + 3 a z + 5 a z + 2 a z + 5 a z + 3 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 142]], Vassiliev[3][Knot[11, Alternating, 142]]}
Out[15]=	{7, -20}
In[16]:=	Kh[Knot[11, Alternating, 142]][q, t]
Out[16]=	2 3 1 1 1 3 1 3 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 7 5 25 9 23 8 21 8 21 7 19 7 19 6 17 6 q q q t q t q t q t q t q t q t 5 3 4 5 4 4 4 4 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 17 5 15 5 15 4 13 4 13 3 11 3 11 2 9 2 q t q t q t q t q t q t q t q t 2 2 4 t t t > ---- + ---- + -- + -- + -- 9 7 5 3 q q t q t q q

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a142

K11a141 K11a143