9 42

9_41

9_43

Visit 9 42's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 42's page at Knotilus! Visit 9 42's page at the original Knot Atlas! 9 42 Quick Notes

9_42 is Alexander Stoimenow's favourite knot!

Alsacian chair, alsacian museum, Strasbourg, France

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 X18,14,1,13 X12,18,13,17
Gauss code	-1, 4, -3, 1, -2, -7, 6, 3, -4, 2, 5, -9, 8, -6, 7, -5, 9, -8
Dowker-Thistlethwaite code	4 8 10 -14 2 -16 -18 -6 -12
Conway Notation	[22,3,2-]

Minimum Braid Representative: Length is 9, width is 4. Braid index is 4.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index 4 Nakanishi index 1 Maximal Thurston-Bennequin number [-3][-5] Hyperbolic Volume 4.05686 A-Polynomial See Data:9 42/A-polynomial

[edit Notes for 9 42's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $2$ Rasmussen s-Invariant 0

[edit Notes for 9 42's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{2}+2t-1+2t^{-1}-t^{-2}$
Conway polynomial	$-z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 7, 2 }
Jones polynomial	$q^{3}-q^{2}+q-1+q^{-1}-q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}+a^{2}z^{2}+z^{2}a^{-2}-4z^{2}+2a^{2}+2a^{-2}-3$
Kauffman polynomial (db, data sources)	$az^{7}+z^{7}a^{-1}+a^{2}z^{6}+z^{6}a^{-2}+2z^{6}-5az^{5}-5z^{5}a^{-1}-5a^{2}z^{4}-5z^{4}a^{-2}-10z^{4}+6az^{3}+6z^{3}a^{-1}+6a^{2}z^{2}+6z^{2}a^{-2}+12z^{2}-2az-2za^{-1}-2a^{2}-2a^{-2}-3$
The A2 invariant	$q^{10}+q^{8}+q^{6}-q^{2}-1-q^{-2}+q^{-6}+q^{-8}+q^{-10}$
The G2 invariant	$q^{46}+q^{42}+2q^{32}+q^{26}+q^{24}+q^{22}+q^{20}-q^{18}+q^{16}+q^{14}-q^{12}+q^{10}-q^{8}-q^{4}-2q^{2}-1-q^{-2}-q^{-4}-2q^{-6}-q^{-8}-q^{-10}+q^{-12}-q^{-14}-q^{-16}+q^{-20}+q^{-22}+q^{-24}+q^{-26}+3q^{-30}+q^{-34}+q^{-36}+q^{-40}+q^{-46}-q^{-50}-q^{-54}+q^{-56}-q^{-60}+q^{-62}$

Further Quantum Invariants

Further quantum knot invariants for 9_42.

A1 Invariants.

Weight	Invariant
1	$q^{7}+q^{-7}$
2	$q^{22}-q^{18}+q^{6}+1+q^{-6}-q^{-18}+q^{-22}$
3	$q^{45}-q^{41}-q^{39}+q^{35}+q^{23}+q^{21}-q^{19}-q^{17}+q^{13}-q^{9}+q^{5}+q^{3}+q^{-3}+q^{-5}-q^{-13}-q^{-15}+q^{-21}+2q^{-23}-q^{-27}+2q^{-31}-3q^{-35}-q^{-37}+q^{-39}+q^{-41}$
4	$q^{76}-q^{72}-q^{70}-q^{68}+q^{66}+q^{64}+q^{62}-q^{58}+q^{48}+q^{46}-2q^{42}-2q^{40}+q^{36}+2q^{34}-q^{30}-q^{28}+2q^{24}+q^{22}+q^{20}-q^{18}-2q^{16}+q^{12}+2q^{10}-2q^{6}-q^{4}+2+q^{-2}-q^{-4}+q^{-8}+2q^{-10}+q^{-12}-q^{-14}+q^{-20}-q^{-24}-q^{-34}-2q^{-36}+q^{-40}+2q^{-42}+2q^{-44}-q^{-46}-q^{-48}-2q^{-50}+q^{-52}+3q^{-54}-2q^{-60}-q^{-62}+q^{-68}$
5	$q^{115}-q^{111}-q^{109}-q^{107}+q^{103}+2q^{101}+q^{99}-q^{95}-q^{93}-q^{91}+q^{87}+q^{81}+q^{79}-q^{75}-3q^{73}-2q^{71}+2q^{67}+3q^{65}+2q^{63}-2q^{59}-2q^{57}-q^{55}+q^{53}+2q^{51}+2q^{49}+q^{47}-q^{45}-2q^{43}-3q^{41}-2q^{39}+q^{37}+3q^{35}+3q^{33}+q^{31}-2q^{29}-4q^{27}-2q^{25}+q^{23}+3q^{21}+4q^{19}+q^{17}-2q^{15}-3q^{13}-q^{11}+2q^{9}+3q^{7}+2q^{5}-q^{3}-3q-2q^{-1}+q^{-3}+3q^{-5}+2q^{-7}-2q^{-11}-q^{-13}+q^{-15}+2q^{-17}+q^{-19}-q^{-21}-q^{-23}+q^{-27}-q^{-31}-q^{-33}+q^{-37}+q^{-39}+q^{-55}+2q^{-57}+q^{-59}-2q^{-61}-4q^{-63}-4q^{-65}+5q^{-69}+6q^{-71}+q^{-73}-5q^{-75}-6q^{-77}-3q^{-79}+3q^{-81}+6q^{-83}+5q^{-85}-3q^{-89}-3q^{-91}-2q^{-93}+q^{-97}+q^{-99}$

A2 Invariants.

Weight	Invariant
1,0	$q^{10}+q^{8}+q^{6}-q^{2}-1-q^{-2}+q^{-6}+q^{-8}+q^{-10}$
1,1	$q^{28}+2q^{24}+2q^{20}-2q^{18}-2q^{16}-2q^{14}-2q^{12}+2q^{6}+4q^{4}+4q^{2}+2+2q^{-2}-2q^{-4}-4q^{-8}-2q^{-12}+2q^{-14}+2q^{-18}+2q^{-24}-2q^{-26}+q^{-28}-2q^{-30}+2q^{-32}$
2,0	$q^{28}+q^{26}+q^{24}-q^{18}-q^{16}-2q^{14}-2q^{12}-q^{10}+q^{8}+3q^{6}+2q^{4}+3q^{2}+2+q^{-2}-q^{-4}-q^{-6}-q^{-8}-q^{-10}+q^{-16}+q^{-20}-q^{-22}-q^{-24}+q^{-28}+q^{-30}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{20}+q^{16}+q^{14}+q^{12}-q^{4}-q^{-4}+q^{-12}+q^{-14}+q^{-16}+q^{-20}$
1,0,0	$q^{13}+q^{11}+2q^{9}+q^{7}-q^{3}-2q-2q^{-1}-q^{-3}+q^{-7}+2q^{-9}+q^{-11}+q^{-13}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{26}+q^{24}+2q^{22}+2q^{20}+2q^{18}+q^{16}-2q^{12}-3q^{10}-3q^{8}-2q^{6}-q^{4}+q^{2}+4+4q^{-2}+3q^{-4}+2q^{-6}-3q^{-10}-3q^{-12}-3q^{-14}-2q^{-16}+2q^{-20}+3q^{-22}+2q^{-24}+2q^{-26}+q^{-28}-q^{-32}$
1,0,0,0	$q^{16}+q^{14}+2q^{12}+2q^{10}+q^{8}-q^{4}-2q^{2}-3-2q^{-2}-q^{-4}+q^{-8}+2q^{-10}+2q^{-12}+q^{-14}+q^{-16}$

B2 Invariants.

Weight	Invariant
0,1	$q^{20}+q^{16}+q^{14}+q^{12}-q^{4}-2-q^{-4}+q^{-12}+q^{-14}+q^{-16}+q^{-20}$
1,0	$q^{34}+q^{26}+q^{18}-q^{14}+1-q^{-14}+q^{-18}+q^{-26}+q^{-34}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{26}+q^{22}+q^{20}+2q^{18}+q^{16}+q^{14}+q^{12}-q^{6}-q^{4}-2q^{2}-2q^{-2}-q^{-4}-q^{-6}+q^{-12}+q^{-14}+q^{-16}+2q^{-18}+q^{-20}+q^{-22}+q^{-26}$

G2 Invariants.

Weight	Invariant
1,0	$q^{46}+q^{42}+2q^{32}+q^{26}+q^{24}+q^{22}+q^{20}-q^{18}+q^{16}+q^{14}-q^{12}+q^{10}-q^{8}-q^{4}-2q^{2}-1-q^{-2}-q^{-4}-2q^{-6}-q^{-8}-q^{-10}+q^{-12}-q^{-14}-q^{-16}+q^{-20}+q^{-22}+q^{-24}+q^{-26}+3q^{-30}+q^{-34}+q^{-36}+q^{-40}+q^{-46}-q^{-50}-q^{-54}+q^{-56}-q^{-60}+q^{-62}$

.

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["9 42"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

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]=

{ 7, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{3}-q^{2}+q-1+q^{-1}-q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{4}+a^{2}z^{2}+z^{2}a^{-2}-4z^{2}+2a^{2}+2a^{-2}-3$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$az^{7}+z^{7}a^{-1}+a^{2}z^{6}+z^{6}a^{-2}+2z^{6}-5az^{5}-5z^{5}a^{-1}-5a^{2}z^{4}-5z^{4}a^{-2}-10z^{4}+6az^{3}+6z^{3}a^{-1}+6a^{2}z^{2}+6z^{2}a^{-2}+12z^{2}-2az-2za^{-1}-2a^{2}-2a^{-2}-3$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {...}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {...}

Vassiliev invariants

V2 and V3:

(-2, 0)

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 $-8$ $0$ $32$ ${\frac {164}{3}}$ ${\frac {76}{3}}$ $0$ $0$ $0$ $0$ $-{\frac {256}{3}}$ $0$ $-{\frac {1312}{3}}$ $-{\frac {608}{3}}$ $-{\frac {6271}{15}}$ ${\frac {1484}{15}}$ $-{\frac {19564}{45}}$ ${\frac {607}{9}}$ $-{\frac {1471}{15}}$

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=$2 is the signature of 9 42. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 χ 7             1 1 5               0 3         1 1   0 1       1 1     0 -1       1 1     0 -3   1 1         0 -5               0 -7 1             1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

$n$	$J_{n}$
2	$q^{10}-q^{9}-q^{8}+2q^{7}-q^{6}-q^{5}+2q^{4}-q^{3}+q-1+q^{-1}-q^{-3}+2q^{-4}-q^{-5}-q^{-6}+2q^{-7}-q^{-8}-q^{-9}+q^{-10}$
3	$q^{19}-2q^{17}-2q^{16}+4q^{15}+2q^{14}-4q^{13}-3q^{12}+5q^{11}+4q^{10}-5q^{9}-4q^{8}+5q^{7}+3q^{6}-5q^{5}-3q^{4}+5q^{3}+3q^{2}-4q-3+4q^{-1}+3q^{-2}-3q^{-3}-3q^{-4}+3q^{-5}+2q^{-6}-2q^{-7}-2q^{-8}+2q^{-9}+q^{-10}-2q^{-11}+2q^{-13}-2q^{-15}+2q^{-17}-q^{-19}-q^{-20}+q^{-21}$
4	$q^{32}-q^{31}-q^{29}-q^{28}+3q^{27}-q^{26}+3q^{25}-3q^{24}-4q^{23}+4q^{22}-q^{21}+6q^{20}-3q^{19}-5q^{18}+3q^{17}-3q^{16}+7q^{15}-2q^{14}-5q^{13}+3q^{12}-3q^{11}+6q^{10}-q^{9}-4q^{8}+2q^{7}-3q^{6}+5q^{5}+q^{4}-3q^{3}+q^{2}-4q+4+3q^{-1}-2q^{-2}-q^{-3}-5q^{-4}+3q^{-5}+5q^{-6}-2q^{-8}-6q^{-9}+q^{-10}+6q^{-11}+2q^{-12}-2q^{-13}-5q^{-14}-q^{-15}+5q^{-16}+2q^{-17}-q^{-18}-3q^{-19}-2q^{-20}+4q^{-21}-q^{-23}-q^{-24}-q^{-25}+4q^{-26}-q^{-27}-q^{-28}-q^{-29}-q^{-30}+3q^{-31}-q^{-34}-q^{-35}+q^{-36}$
5	$q^{47}-q^{45}-2q^{44}-q^{43}+4q^{41}+5q^{40}-5q^{38}-7q^{37}-3q^{36}+5q^{35}+11q^{34}+5q^{33}-6q^{32}-12q^{31}-7q^{30}+5q^{29}+13q^{28}+8q^{27}-5q^{26}-13q^{25}-8q^{24}+5q^{23}+13q^{22}+8q^{21}-5q^{20}-13q^{19}-8q^{18}+6q^{17}+13q^{16}+7q^{15}-6q^{14}-13q^{13}-7q^{12}+7q^{11}+12q^{10}+6q^{9}-6q^{8}-11q^{7}-6q^{6}+6q^{5}+10q^{4}+5q^{3}-4q^{2}-9q-5+4q^{-1}+7q^{-2}+4q^{-3}-2q^{-4}-6q^{-5}-4q^{-6}+3q^{-7}+4q^{-8}+2q^{-9}-q^{-10}-3q^{-11}-q^{-12}+2q^{-13}+2q^{-14}-q^{-15}-3q^{-16}-q^{-17}+2q^{-18}+4q^{-19}+2q^{-20}-3q^{-21}-6q^{-22}-2q^{-23}+3q^{-24}+5q^{-25}+4q^{-26}-2q^{-27}-6q^{-28}-3q^{-29}+q^{-30}+4q^{-31}+4q^{-32}-4q^{-34}-2q^{-35}+2q^{-37}+2q^{-38}-q^{-39}-2q^{-40}-q^{-41}+q^{-42}+2q^{-43}+q^{-44}-q^{-45}-q^{-46}-2q^{-47}+2q^{-49}+q^{-50}-q^{-53}-q^{-54}+q^{-55}$
6	$q^{66}-q^{65}-q^{62}-q^{61}-q^{60}+3q^{59}+q^{58}+4q^{57}+q^{56}-2q^{55}-7q^{54}-6q^{53}+2q^{51}+14q^{50}+7q^{49}+2q^{48}-13q^{47}-14q^{46}-7q^{45}-q^{44}+20q^{43}+13q^{42}+9q^{41}-12q^{40}-16q^{39}-13q^{38}-5q^{37}+20q^{36}+14q^{35}+13q^{34}-11q^{33}-14q^{32}-15q^{31}-7q^{30}+19q^{29}+13q^{28}+14q^{27}-11q^{26}-14q^{25}-15q^{24}-6q^{23}+19q^{22}+13q^{21}+14q^{20}-10q^{19}-14q^{18}-15q^{17}-5q^{16}+17q^{15}+11q^{14}+14q^{13}-8q^{12}-12q^{11}-15q^{10}-5q^{9}+13q^{8}+9q^{7}+16q^{6}-5q^{5}-9q^{4}-16q^{3}-6q^{2}+8q+7+18q^{-1}-5q^{-3}-17q^{-4}-8q^{-5}+2q^{-6}+4q^{-7}+19q^{-8}+5q^{-9}-16q^{-11}-8q^{-12}-4q^{-13}-q^{-14}+17q^{-15}+8q^{-16}+4q^{-17}-12q^{-18}-5q^{-19}-7q^{-20}-5q^{-21}+12q^{-22}+7q^{-23}+5q^{-24}-8q^{-25}-5q^{-27}-5q^{-28}+8q^{-29}+3q^{-30}+q^{-31}-7q^{-32}+2q^{-33}-q^{-34}-q^{-35}+8q^{-36}+2q^{-37}-3q^{-38}-8q^{-39}-q^{-40}-q^{-41}+q^{-42}+9q^{-43}+4q^{-44}-2q^{-45}-6q^{-46}-3q^{-47}-3q^{-48}-q^{-49}+8q^{-50}+3q^{-51}-2q^{-53}-2q^{-54}-2q^{-55}-2q^{-56}+6q^{-57}-q^{-59}-q^{-60}-q^{-61}-q^{-62}-q^{-63}+5q^{-64}-q^{-67}-q^{-68}-2q^{-69}-q^{-70}+3q^{-71}+q^{-73}-q^{-76}-q^{-77}+q^{-78}$
7	$q^{87}-q^{85}-q^{84}-q^{83}-q^{82}+q^{80}+5q^{79}+4q^{78}+q^{77}-q^{76}-5q^{75}-7q^{74}-9q^{73}-4q^{72}+7q^{71}+15q^{70}+11q^{69}+9q^{68}-q^{67}-15q^{66}-22q^{65}-20q^{64}+q^{63}+18q^{62}+23q^{61}+24q^{60}+8q^{59}-15q^{58}-27q^{57}-30q^{56}-10q^{55}+15q^{54}+25q^{53}+30q^{52}+13q^{51}-12q^{50}-24q^{49}-31q^{48}-13q^{47}+12q^{46}+23q^{45}+29q^{44}+13q^{43}-13q^{42}-22q^{41}-29q^{40}-12q^{39}+13q^{38}+22q^{37}+29q^{36}+12q^{35}-13q^{34}-23q^{33}-29q^{32}-11q^{31}+14q^{30}+23q^{29}+28q^{28}+11q^{27}-14q^{26}-24q^{25}-27q^{24}-9q^{23}+15q^{22}+22q^{21}+25q^{20}+9q^{19}-14q^{18}-22q^{17}-24q^{16}-7q^{15}+13q^{14}+19q^{13}+21q^{12}+8q^{11}-11q^{10}-17q^{9}-19q^{8}-7q^{7}+9q^{6}+13q^{5}+16q^{4}+9q^{3}-6q^{2}-10q-14-7q^{-1}+3q^{-2}+5q^{-3}+10q^{-4}+9q^{-5}-3q^{-7}-6q^{-8}-6q^{-9}-2q^{-10}-3q^{-11}+q^{-12}+6q^{-13}+3q^{-14}+5q^{-15}+3q^{-16}-q^{-17}-2q^{-18}-8q^{-19}-9q^{-20}-2q^{-21}+q^{-22}+8q^{-23}+11q^{-24}+7q^{-25}+4q^{-26}-7q^{-27}-14q^{-28}-10q^{-29}-6q^{-30}+3q^{-31}+12q^{-32}+13q^{-33}+10q^{-34}-11q^{-36}-11q^{-37}-11q^{-38}-4q^{-39}+7q^{-40}+10q^{-41}+10q^{-42}+4q^{-43}-4q^{-44}-6q^{-45}-7q^{-46}-5q^{-47}+3q^{-48}+5q^{-49}+4q^{-50}+q^{-51}-3q^{-52}-3q^{-53}-2q^{-54}+4q^{-56}+6q^{-57}+2q^{-58}-2q^{-59}-6q^{-60}-6q^{-61}-2q^{-62}+4q^{-64}+8q^{-65}+5q^{-66}-4q^{-68}-7q^{-69}-3q^{-70}-2q^{-71}+6q^{-73}+4q^{-74}+2q^{-75}-q^{-76}-4q^{-77}-q^{-78}-q^{-79}-q^{-80}+4q^{-81}+q^{-82}-3q^{-85}-q^{-86}-q^{-87}+4q^{-89}+q^{-90}+q^{-92}-2q^{-93}-q^{-94}-2q^{-95}-q^{-96}+2q^{-97}+q^{-98}+q^{-100}-q^{-103}-q^{-104}+q^{-105}$

Computer Talk

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

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[9, 42]]
Out[2]=	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[18, 14, 1, 13], X[12, 18, 13, 17]]
In[3]:=	GaussCode[Knot[9, 42]]
Out[3]=	GaussCode[-1, 4, -3, 1, -2, -7, 6, 3, -4, 2, 5, -9, 8, -6, 7, -5, 9, -8]
In[4]:=	DTCode[Knot[9, 42]]
Out[4]=	DTCode[4, 8, 10, -14, 2, -16, -18, -6, -12]
In[5]:=	br = BR[Knot[9, 42]]
Out[5]=	BR[4, {1, 1, 1, -2, -1, -1, 3, -2, 3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 9}
In[7]:=	BraidIndex[Knot[9, 42]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[9, 42]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[9, 42]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 2, 3, 4, 1}
In[10]:=	alex = Alexander[Knot[9, 42]][t]
Out[10]=	-2 2 2 -1 - t + - + 2 t - t t
In[11]:=	Conway[Knot[9, 42]][z]
Out[11]=	2 4 1 - 2 z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 42]}
In[13]:=	{KnotDet[Knot[9, 42]], KnotSignature[Knot[9, 42]]}
Out[13]=	{7, 2}
In[14]:=	Jones[Knot[9, 42]][q]
Out[14]=	-3 -2 1 2 3 -1 + q - q + - + q - q + q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 42]}
In[16]:=	A2Invariant[Knot[9, 42]][q]
Out[16]=	-10 -8 -6 -2 2 6 8 10 -1 + q + q + q - q - q + q + q + q
In[17]:=	HOMFLYPT[Knot[9, 42]][a, z]
Out[17]=	2 2 2 2 z 2 2 4 -3 + -- + 2 a - 4 z + -- + a z - z 2 2 a a
In[18]:=	Kauffman[Knot[9, 42]][a, z]
Out[18]=	2 3 2 2 2 z 2 6 z 2 2 6 z 3 -3 - -- - 2 a - --- - 2 a z + 12 z + ---- + 6 a z + ---- + 6 a z - 2 a 2 a a a 4 5 6 7 4 5 z 2 4 5 z 5 6 z 2 6 z 7 10 z - ---- - 5 a z - ---- - 5 a z + 2 z + -- + a z + -- + a z 2 a 2 a a a
In[19]:=	{Vassiliev[2][Knot[9, 42]], Vassiliev[3][Knot[9, 42]]}
Out[19]=	{-2, 0}
In[20]:=	Kh[Knot[9, 42]][q, t]
Out[20]=	1 3 1 1 1 1 q 3 7 2 - + q + q + ----- + ----- + ----- + --- + - + q t + q t q 7 4 3 3 3 2 q t t q t q t q t
In[21]:=	ColouredJones[Knot[9, 42], 2][q]
Out[21]=	-10 -9 -8 2 -6 -5 2 -3 1 3 4 -1 + q - q - q + -- - q - q + -- - q + - + q - q + 2 q - 7 4 q q q 5 6 7 8 9 10 q - q + 2 q - q - q + q

See/edit the Rolfsen_Splice_Template.