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

K11n41 K11n43

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X12,5,13,6 X2837 X9,18,10,19 X11,21,12,20 X6,13,7,14 X15,10,16,11 X17,22,18,1 X19,15,20,14 X21,16,22,17

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

1

Conway Polynomial:

1

Other knots with the same Alexander/Conway Polynomial:

{01, K11n34, ...}

Determinant and Signature:

{1, 0}

Jones Polynomial:

q-6 - 2q-5 + 2q-4 - 2q-3 + q-2 + 2q - 2q2 + 2q3 - q4

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

{K11n34, ...}

A2 (sl(3)) Invariant:

q-18 + q-14 - q-12 - q-10 - q-8 - 2q-6 + q-4 + 3 + 2q2 + q4 + q6 - q8 - q12

HOMFLY-PT Polynomial:

- 2a-2 - 3a-2z2 - a-2z4 + 7 + 11z2 + 6z4 + z6 - 6a2 - 11a2z2 - 6a2z4 - a2z6 + 2a4 + 3a4z2 + a4z4

Kauffman Polynomial:

- 2a-3z + 6a-3z3 - 5a-3z5 + a-3z7 + 2a-2 - 9a-2z2 + 16a-2z4 - 11a-2z6 + 2a-2z8 - 5a-1z + 11a-1z3 - 2a-1z5 - 4a-1z7 + a-1z9 + 7 - 24z2 + 36z4 - 20z6 + 3z8 - 7az + 12az3 - 5az7 + az9 + 6a2 - 20a2z2 + 26a2z4 - 14a2z6 + 2a2z8 - 7a3z + 16a3z3 - 12a3z5 + 2a3z7 + 2a4 - 2a4z2 + 2a4z4 - 4a4z6 + a4z8 - 3a5z + 9a5z3 - 9a5z5 + 2a5z7 + 3a6z2 - 4a6z4 + a6z6

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

{0, 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=0 is the signature of 1142. 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                     1   j = 5                   1 1   j = 3               1 2 1     j = 1             2 1 1       j = -1           1 3 2         j = -3         2 2 1           j = -5       1 1 1             j = -7     1 2 1               j = -9   1 1                   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, 42]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 42]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[12, 5, 13, 6], X[2, 8, 3, 7], > X[9, 18, 10, 19], X[11, 21, 12, 20], X[6, 13, 7, 14], X[15, 10, 16, 11], > X[17, 22, 18, 1], X[19, 15, 20, 14], X[21, 16, 22, 17]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 42]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -7, 4, -2, -5, 8, -6, -3, 7, 10, -8, 11, -9, 5, -10, > 6, -11, 9]
In[5]:=	DTCode[Knot[11, NonAlternating, 42]]
Out[5]=	DTCode[4, 8, 12, 2, -18, -20, 6, -10, -22, -14, -16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 42]][t]
Out[6]=	1
In[7]:=	Conway[Knot[11, NonAlternating, 42]][z]
Out[7]=	1
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[0, 1], Knot[11, NonAlternating, 34], Knot[11, NonAlternating, 42]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 42]], KnotSignature[Knot[11, NonAlternating, 42]]}
Out[9]=	{1, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 42]][q]
Out[10]=	-6 2 2 2 -2 2 3 4 q - -- + -- - -- + q + 2 q - 2 q + 2 q - q 5 4 3 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 34], Knot[11, NonAlternating, 42]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 42]][q]
Out[12]=	-18 -14 -12 -10 -8 2 -4 2 4 6 8 12 3 + q + q - q - q - q - -- + q + 2 q + q + q - q - q 6 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 42]][a, z]
Out[13]=	2 4 2 2 4 2 3 z 2 2 4 2 4 z 7 - -- - 6 a + 2 a + 11 z - ---- - 11 a z + 3 a z + 6 z - -- - 2 2 2 a a a 2 4 4 4 6 2 6 > 6 a z + a z + z - a z
In[14]:=	Kauffman[Knot[11, NonAlternating, 42]][a, z]
Out[14]=	2 2 2 4 2 z 5 z 3 5 2 9 z 7 + -- + 6 a + 2 a - --- - --- - 7 a z - 7 a z - 3 a z - 24 z - ---- - 2 3 a 2 a a a 3 3 2 2 4 2 6 2 6 z 11 z 3 3 3 > 20 a z - 2 a z + 3 a z + ---- + ----- + 12 a z + 16 a z + 3 a a 4 5 5 5 3 4 16 z 2 4 4 4 6 4 5 z 2 z > 9 a z + 36 z + ----- + 26 a z + 2 a z - 4 a z - ---- - ---- - 2 3 a a a 6 7 3 5 5 5 6 11 z 2 6 4 6 6 6 z > 12 a z - 9 a z - 20 z - ----- - 14 a z - 4 a z + a z + -- - 2 3 a a 7 8 9 4 z 7 3 7 5 7 8 2 z 2 8 4 8 z > ---- - 5 a z + 2 a z + 2 a z + 3 z + ---- + 2 a z + a z + -- + a 2 a a 9 > a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 42]], Vassiliev[3][Knot[11, NonAlternating, 42]]}
Out[15]=	{0, 2}
In[16]:=	Kh[Knot[11, NonAlternating, 42]][q, t]
Out[16]=	-3 3 1 1 1 1 1 2 1 q + - + 2 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 q t q t q t q t q t q t q t 1 1 2 1 2 1 2 t 3 2 > ----- + ----- + ----- + ---- + ---- + --- + --- + q t + q t + q t + 7 2 5 2 3 2 5 3 q t q q t q t q t q t q t 3 2 3 3 5 3 5 4 7 4 9 5 > 2 q t + q t + q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n42

K11n41 K11n43