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

K11a140 K11a142

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 16 18 12 2 8 20 22 14 6

Alexander Polynomial:

2t-3 - 11t-2 + 24t-1 - 29 + 24t - 11t2 + 2t3

Conway Polynomial:

1 - 2z2 + z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a12, ...}

Determinant and Signature:

{103, -2}

Jones Polynomial:

q-7 - 3q-6 + 6q-5 - 10q-4 + 14q-3 - 16q-2 + 16q-1 - 14 + 11q - 7q2 + 4q3 - q4

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

{...}

A2 (sl(3)) Invariant:

q-22 - q-18 + 2q-16 - 2q-14 - q-12 + 2q-10 - 2q-8 + 3q-6 - q-4 + 1 - 3q2 + 3q4 + 2q10 - q12

HOMFLY-PT Polynomial:

a-2 - a-2z2 - a-2z4 - 1 + 2z4 + z6 + 2a2 + 2a2z2 + 2a2z4 + a2z6 - 2a4 - 4a4z2 - 2a4z4 + a6 + a6z2

Kauffman Polynomial:

2a-3z3 - 3a-3z5 + a-3z7 - a-2 - 3a-2z2 + 15a-2z4 - 15a-2z6 + 4a-2z8 - 2a-1z3 + 14a-1z5 - 17a-1z7 + 5a-1z9 - 1 - 6z2 + 30z4 - 28z6 + 3z8 + 2z10 + az - 7az3 + 24az5 - 30az7 + 10az9 - 2a2 + 16a2z4 - 23a2z6 + 5a2z8 + 2a2z10 + a3z - a3z5 - 6a3z7 + 5a3z9 - 2a4 + 7a4z2 - 5a4z4 - 5a4z6 + 6a4z8 + a5z - 5a5z5 + 6a5z7 - a6 + 3a6z2 - 5a6z4 + 5a6z6 + a7z - 3a7z3 + 3a7z5 - a8z2 + a8z4

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

{-2, 2}

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 11141. 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                     3   j = 5                   4 1   j = 3                 7 3     j = 1               7 4       j = -1             9 7         j = -3           8 8           j = -5         6 8             j = -7       4 8               j = -9     2 6                 j = -11   1 4                   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, 141]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 141]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[16, 6, 17, 5], X[18, 7, 19, 8], > X[12, 10, 13, 9], X[2, 11, 3, 12], X[8, 14, 9, 13], X[20, 15, 21, 16], > X[22, 18, 1, 17], X[14, 19, 15, 20], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 141]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -11, 4, -7, 5, -2, 6, -5, 7, -10, 8, -3, 9, -4, 10, > -8, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 141]]
Out[5]=	DTCode[4, 10, 16, 18, 12, 2, 8, 20, 22, 14, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 141]][t]
Out[6]=	2 11 24 2 3 -29 + -- - -- + -- + 24 t - 11 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 141]][z]
Out[7]=	2 4 6 1 - 2 z + z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 12], Knot[11, Alternating, 141]}
In[9]:=	{KnotDet[Knot[11, Alternating, 141]], KnotSignature[Knot[11, Alternating, 141]]}
Out[9]=	{103, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 141]][q]
Out[10]=	-7 3 6 10 14 16 16 2 3 4 -14 + q - -- + -- - -- + -- - -- + -- + 11 q - 7 q + 4 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, 141]}
In[12]:=	A2Invariant[Knot[11, Alternating, 141]][q]
Out[12]=	-22 -18 2 2 -12 2 2 3 -4 2 4 1 + q - q + --- - --- - q + --- - -- + -- - q - 3 q + 3 q + 16 14 10 8 6 q q q q q 10 12 > 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 141]][a, z]
Out[13]=	2 4 -2 2 4 6 z 2 2 4 2 6 2 4 z -1 + a + 2 a - 2 a + a - -- + 2 a z - 4 a z + a z + 2 z - -- + 2 2 a a 2 4 4 4 6 2 6 > 2 a z - 2 a z + z + a z
In[14]:=	Kauffman[Knot[11, Alternating, 141]][a, z]
Out[14]=	2 -2 2 4 6 3 5 7 2 3 z -1 - a - 2 a - 2 a - a + a z + a z + a z + a z - 6 z - ---- + 2 a 3 3 4 2 6 2 8 2 2 z 2 z 3 7 3 4 > 7 a z + 3 a z - a z + ---- - ---- - 7 a z - 3 a z + 30 z + 3 a a 4 5 5 15 z 2 4 4 4 6 4 8 4 3 z 14 z 5 > ----- + 16 a z - 5 a z - 5 a z + a z - ---- + ----- + 24 a z - 2 3 a a a 6 3 5 5 5 7 5 6 15 z 2 6 4 6 6 6 > a z - 5 a z + 3 a z - 28 z - ----- - 23 a z - 5 a z + 5 a z + 2 a 7 7 8 z 17 z 7 3 7 5 7 8 4 z 2 8 > -- - ----- - 30 a z - 6 a z + 6 a z + 3 z + ---- + 5 a z + 3 a 2 a a 9 4 8 5 z 9 3 9 10 2 10 > 6 a z + ---- + 10 a z + 5 a z + 2 z + 2 a z a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 141]], Vassiliev[3][Knot[11, Alternating, 141]]}
Out[15]=	{-2, 2}
In[16]:=	Kh[Knot[11, Alternating, 141]][q, t]
Out[16]=	8 9 1 2 1 4 2 6 4 8 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 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 6 8 8 7 t 2 3 2 3 3 5 3 > ----- + ---- + ---- + --- + 7 q t + 4 q t + 7 q t + 3 q t + 4 q t + 5 2 5 3 q q t q t q t 5 4 7 4 9 5 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a141

K11a140 K11a142