10 137

10_136

10_138

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10 137 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X5,10,6,11 X3948 X9,3,10,2 X16,12,17,11 X14,7,15,8 X6,15,7,16 X20,18,1,17 X18,13,19,14 X12,19,13,20
Gauss code	-1, 4, -3, 1, -2, -7, 6, 3, -4, 2, 5, -10, 9, -6, 7, -5, 8, -9, 10, -8
Dowker-Thistlethwaite code	4 8 10 -14 2 -16 -18 -6 -20 -12
Conway Notation	[22,211,2-]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-7][-3] Hyperbolic Volume 9.25056 A-Polynomial See Data:10 137/A-polynomial

[edit Notes for 10 137's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $0$ Topological 4 genus $0$ Concordance genus $0$ Rasmussen s-Invariant 0

[edit Notes for 10 137's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{2}-6t+11-6t^{-1}+t^{-2}$
Conway polynomial	$z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 25, 0 }
Jones polynomial	$q^{2}-2q+4-4q^{-1}+4q^{-2}-4q^{-3}+3q^{-4}-2q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$a^{6}-2z^{2}a^{4}-2a^{4}+z^{4}a^{2}+2z^{2}a^{2}+2a^{2}-2z^{2}-1+a^{-2}$
Kauffman polynomial (db, data sources)	$a^{4}z^{8}+a^{2}z^{8}+2a^{5}z^{7}+4a^{3}z^{7}+2az^{7}+a^{6}z^{6}-a^{4}z^{6}-a^{2}z^{6}+z^{6}-8a^{5}z^{5}-15a^{3}z^{5}-7az^{5}-4a^{6}z^{4}-7a^{4}z^{4}-5a^{2}z^{4}-2z^{4}+8a^{5}z^{3}+15a^{3}z^{3}+9az^{3}+2z^{3}a^{-1}+4a^{6}z^{2}+8a^{4}z^{2}+7a^{2}z^{2}+z^{2}a^{-2}+4z^{2}-3a^{5}z-5a^{3}z-3az-za^{-1}-a^{6}-2a^{4}-2a^{2}-a^{-2}-1$
The A2 invariant	$q^{20}+q^{18}-q^{16}-q^{12}-q^{10}+q^{8}+q^{4}+q^{-2}-q^{-4}+q^{-6}+q^{-8}$
The G2 invariant	$q^{94}-q^{92}+3q^{90}-4q^{88}+3q^{86}-q^{84}-4q^{82}+10q^{80}-10q^{78}+10q^{76}-4q^{74}-4q^{72}+11q^{70}-12q^{68}+8q^{66}-q^{64}-6q^{62}+8q^{60}-6q^{58}-2q^{56}+9q^{54}-13q^{52}+10q^{50}-5q^{48}-6q^{46}+12q^{44}-15q^{42}+13q^{40}-9q^{38}+3q^{36}+5q^{34}-9q^{32}+11q^{30}-9q^{28}+5q^{26}+3q^{24}-6q^{22}+6q^{20}-2q^{18}-3q^{16}+10q^{14}-11q^{12}+7q^{10}+q^{8}-10q^{6}+14q^{4}-13q^{2}+7+q^{-2}-7q^{-4}+7q^{-6}-5q^{-8}+3q^{-10}+q^{-12}-2q^{-14}+q^{-16}-2q^{-20}+3q^{-22}+2q^{-28}-q^{-30}+q^{-32}+q^{-38}$

Further Quantum Invariants

Further quantum knot invariants for 10_137.

A1 Invariants.

Weight	Invariant
1	$q^{13}-q^{11}+q^{9}-q^{7}+2q^{-1}-q^{-3}+q^{-5}$
2	$q^{38}-q^{36}-2q^{34}+3q^{32}+q^{30}-4q^{28}+q^{26}+3q^{24}-2q^{22}-2q^{20}+3q^{18}+q^{16}-3q^{14}+2q^{12}+2q^{10}-3q^{8}-q^{6}+3q^{4}-3+3q^{-2}+2q^{-4}-3q^{-6}+2q^{-10}$
3	$q^{75}-q^{73}-2q^{71}+4q^{67}+3q^{65}-5q^{63}-6q^{61}+2q^{59}+8q^{57}+3q^{55}-7q^{53}-7q^{51}+2q^{49}+10q^{47}+5q^{45}-10q^{43}-10q^{41}+5q^{39}+14q^{37}-2q^{35}-15q^{33}-q^{31}+14q^{29}+4q^{27}-12q^{25}-4q^{23}+11q^{21}+5q^{19}-10q^{17}-5q^{15}+6q^{13}+7q^{11}-3q^{9}-8q^{7}-4q^{5}+10q^{3}+11q-5q^{-1}-15q^{-3}+18q^{-7}+3q^{-9}-14q^{-11}-7q^{-13}+9q^{-15}+8q^{-17}-4q^{-19}-5q^{-21}+q^{-23}+2q^{-25}+q^{-27}-q^{-29}$

A2 Invariants.

Weight	Invariant
1,0	$q^{20}+q^{18}-q^{16}-q^{12}-q^{10}+q^{8}+q^{4}+q^{-2}-q^{-4}+q^{-6}+q^{-8}$
2,0	$q^{52}+q^{50}-3q^{46}-2q^{44}+q^{42}+2q^{40}-q^{36}+2q^{34}+2q^{32}-3q^{28}-q^{26}+q^{22}+q^{18}+3q^{16}-2q^{10}-q^{8}+q^{4}-q^{2}-1+2q^{-2}+3q^{-4}-3q^{-8}+q^{-10}+2q^{-12}+q^{-20}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{40}-q^{38}+q^{36}+q^{34}-2q^{32}+q^{30}-q^{28}-q^{26}+2q^{24}+2q^{18}-q^{14}-q^{10}-q^{6}+2q^{2}-1+q^{-2}+2q^{-4}-2q^{-6}+q^{-8}+2q^{-10}+q^{-16}$
1,0,0	$q^{27}+q^{25}+q^{23}-q^{21}-2q^{17}-q^{15}-q^{13}+q^{11}+q^{9}+q^{7}+q^{5}-q^{-1}+q^{-3}-q^{-5}+q^{-7}+q^{-9}+q^{-11}$

B2 Invariants.

Weight	Invariant
0,1	$q^{40}-q^{38}+3q^{36}-3q^{34}+4q^{32}-3q^{30}+3q^{28}-3q^{26}-4q^{20}+4q^{18}-6q^{16}+7q^{14}-6q^{12}+7q^{10}-4q^{8}+3q^{6}+1-3q^{-2}+4q^{-4}-4q^{-6}+3q^{-8}-2q^{-10}+2q^{-12}+q^{-16}$
1,0	$q^{66}-q^{62}-q^{60}+2q^{58}+2q^{56}-2q^{54}-3q^{52}+q^{50}+3q^{48}-4q^{44}-2q^{42}+4q^{40}+3q^{38}-q^{36}-3q^{34}+q^{32}+3q^{30}+q^{28}-2q^{26}-q^{24}+2q^{22}+q^{20}-2q^{18}-3q^{16}+q^{14}+3q^{12}-q^{10}-4q^{8}+3q^{4}+2q^{2}-2-2q^{-2}+2q^{-4}+4q^{-6}-q^{-8}-3q^{-10}+2q^{-14}+2q^{-16}-q^{-20}+q^{-26}$

G2 Invariants.

Weight

Invariant

1,0

$q^{94}-q^{92}+3q^{90}-4q^{88}+3q^{86}-q^{84}-4q^{82}+10q^{80}-10q^{78}+10q^{76}-4q^{74}-4q^{72}+11q^{70}-12q^{68}+8q^{66}-q^{64}-6q^{62}+8q^{60}-6q^{58}-2q^{56}+9q^{54}-13q^{52}+10q^{50}-5q^{48}-6q^{46}+12q^{44}-15q^{42}+13q^{40}-9q^{38}+3q^{36}+5q^{34}-9q^{32}+11q^{30}-9q^{28}+5q^{26}+3q^{24}-6q^{22}+6q^{20}-2q^{18}-3q^{16}+10q^{14}-11q^{12}+7q^{10}+q^{8}-10q^{6}+14q^{4}-13q^{2}+7+q^{-2}-7q^{-4}+7q^{-6}-5q^{-8}+3q^{-10}+q^{-12}-2q^{-14}+q^{-16}-2q^{-20}+3q^{-22}+2q^{-28}-q^{-30}+q^{-32}+q^{-38}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 137"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$t^{2}-6t+11-6t^{-1}+t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{4}-2z^{2}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 25, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$q^{2}-2q+4-4q^{-1}+4q^{-2}-4q^{-3}+3q^{-4}-2q^{-5}+q^{-6}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$a^{6}-2z^{2}a^{4}-2a^{4}+z^{4}a^{2}+2z^{2}a^{2}+2a^{2}-2z^{2}-1+a^{-2}$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$a^{4}z^{8}+a^{2}z^{8}+2a^{5}z^{7}+4a^{3}z^{7}+2az^{7}+a^{6}z^{6}-a^{4}z^{6}-a^{2}z^{6}+z^{6}-8a^{5}z^{5}-15a^{3}z^{5}-7az^{5}-4a^{6}z^{4}-7a^{4}z^{4}-5a^{2}z^{4}-2z^{4}+8a^{5}z^{3}+15a^{3}z^{3}+9az^{3}+2z^{3}a^{-1}+4a^{6}z^{2}+8a^{4}z^{2}+7a^{2}z^{2}+z^{2}a^{-2}+4z^{2}-3a^{5}z-5a^{3}z-3az-za^{-1}-a^{6}-2a^{4}-2a^{2}-a^{-2}-1$

Vassiliev invariants

V2 and V3:

(-2, 2)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $-8$ $16$ $32$ ${\frac {20}{3}}$ ${\frac {28}{3}}$ $-128$ $-{\frac {416}{3}}$ $-{\frac {128}{3}}$ $-16$ $-{\frac {256}{3}}$ $128$ $-{\frac {160}{3}}$ $-{\frac {224}{3}}$ ${\frac {4409}{15}}$ ${\frac {308}{5}}$ ${\frac {1556}{45}}$ ${\frac {151}{9}}$ $-{\frac {151}{15}}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). 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 10 137. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 χ 5                 1 1 3               1   -1 1             3 1   2 -1           2 2     0 -3         2 2       0 -5       2 2         0 -7     1 2           -1 -9   1 2             1 -11   1               -1 -13 1                 1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[10, 137]]
Out[2]=	10
In[3]:=	PD[Knot[10, 137]]
Out[3]=	PD[X[1, 4, 2, 5], X[5, 10, 6, 11], X[3, 9, 4, 8], X[9, 3, 10, 2], X[16, 12, 17, 11], X[14, 7, 15, 8], X[6, 15, 7, 16], X[20, 18, 1, 17], X[18, 13, 19, 14], X[12, 19, 13, 20]]
In[4]:=	GaussCode[Knot[10, 137]]
Out[4]=	GaussCode[-1, 4, -3, 1, -2, -7, 6, 3, -4, 2, 5, -10, 9, -6, 7, -5, 8, -9, 10, -8]
In[5]:=	BR[Knot[10, 137]]
Out[5]=	BR[5, {-1, 2, -1, 2, -3, -2, -2, 4, -3, 4}]
In[6]:=	alex = Alexander[Knot[10, 137]][t]
Out[6]=	-2 6 2 11 + t - - - 6 t + t t
In[7]:=	Conway[Knot[10, 137]][z]
Out[7]=	2 4 1 - 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 137]}
In[9]:=	{KnotDet[Knot[10, 137]], KnotSignature[Knot[10, 137]]}
Out[9]=	{25, 0}
In[10]:=	J=Jones[Knot[10, 137]][q]
Out[10]=	-6 2 3 4 4 4 2 4 + q - -- + -- - -- + -- - - - 2 q + 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[10, 137], Knot[10, 155], Knot[11, NonAlternating, 37]}
In[12]:=	A2Invariant[Knot[10, 137]][q]
Out[12]=	-20 -18 -16 -12 -10 -8 -4 2 4 6 8 q + q - q - q - q + q + q + q - q + q + q
In[13]:=	Kauffman[Knot[10, 137]][a, z]
Out[13]=	2 -2 2 4 6 z 3 5 2 z -1 - a - 2 a - 2 a - a - - - 3 a z - 5 a z - 3 a z + 4 z + -- + a 2 a 3 2 2 4 2 6 2 2 z 3 3 3 5 3 7 a z + 8 a z + 4 a z + ---- + 9 a z + 15 a z + 8 a z - a 4 2 4 4 4 6 4 5 3 5 5 5 2 z - 5 a z - 7 a z - 4 a z - 7 a z - 15 a z - 8 a z + 6 2 6 4 6 6 6 7 3 7 5 7 2 8 z - a z - a z + a z + 2 a z + 4 a z + 2 a z + a z + 4 8 a z
In[14]:=	{Vassiliev[2][Knot[10, 137]], Vassiliev[3][Knot[10, 137]]}
Out[14]=	{0, 2}
In[15]:=	Kh[Knot[10, 137]][q, t]
Out[15]=	2 1 1 1 2 1 2 2 - + 3 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 2 2 2 2 3 5 2 ----- + ----- + ---- + --- + q t + q t + q t 5 2 3 2 3 q t q t q t q t